For a deeper understanding of the relationships between "Firm

Keywords: Digitalization, Supply Chain, Challenges, Digital Tools, Firm Size, Resource-Based View

In a context of unprecedented crises and uncertainties, digital tools appear to bring more visibility to decision-takers along the Supply Chain (SC). They enhance information sharing and collaboration between SC entities. This research contributes to providing a more complete understanding of the relationships between the size of the firm and its adoption of SC digital tools and challenges encountered. For this purpose, we mobilize the Resource-Based View (RBV) and analyze 311 surveys collected from SC professionals using the Kruskal-Wallis and Mann-Whitney tests. Our results clarify the role of firm size in SC digitalization. More precisely, groups of different firm sizes show a significant difference of SC digital tools adoption level and our study enables researchers and practitioners to understand that this difference depends on the type of tools (either operational or support tools). In addition, this study brings a counter-intuitive result that is not consistent with previous research: there is no difference between the groups of different firm sizes with regards to SC digitalization challenges encountered.

Introduction

Our modern business world presents numerous challenges for Supply Chains (SCs).

Financial, economic, social, ecological but also sanitary crises remind firms on a daily basis how uncertain their environments are and that disruptions may occur [START_REF] Dwivedi | Impact of COVID-19 pandemic on information management research and practice: Transforming education, work and life[END_REF][START_REF] Ivanov | Coronavirus (COVID-19/SARS-CoV-2) and supply chain resilience: a research note[END_REF][START_REF] Ruel | Disaster readiness' influence on the impact of supply chain resilience and robustness on firms' financial performance: a COVID-19 empirical investigation[END_REF]. In this context, digital tools appear to be solutions (Ivanov and Dolgui, 2019;Ivanov, 2021a): they bring more visibility to decisiontakers along with the SC due to the collection of data in real-time, enhance information sharing and collaboration between SC entities, and automatically analyze more data [START_REF] Kache | Challenges and opportunities of digital information at the intersection of Big Data Analytics and supply chain management[END_REF][START_REF] Min | Blockchain technology for enhancing supply chain resilience[END_REF]. Therefore, many studies show, by examining certain specific SC digital tools at a time (e.g., [START_REF] Min | Blockchain technology for enhancing supply chain resilience[END_REF][START_REF] Dubey | Empirical investigation of data analytics capability and organizational flexibility as complements to supply chain resilience[END_REF], the various advantages of adopting SC digital tools. However, researchers have indicated that the implementation of digital tools is a complex process, and firms face numerous difficulties [START_REF] Frank | Industry 4.0 technologies: Implementation patterns in manufacturing companies[END_REF][START_REF] Kamble | Analysis of the driving and dependence power of barriers to adopt industry 4.0 in Indian manufacturing industry[END_REF]. The main reasons for the complexity lie in the high investment levels and the unclear return on investment of digital tools [START_REF] Kamble | Analysis of the driving and dependence power of barriers to adopt industry 4.0 in Indian manufacturing industry[END_REF]. Indeed, the literature provides many studies concerning the barriers and obstacles that inhibit the implementation of new technologies. For instance, front liner barriers for SC digitalization are those challenges relating infrastructural issues, absence of adequately skilled workforce and lack of standards [START_REF] Frank | Industry 4.0 technologies: Implementation patterns in manufacturing companies[END_REF][START_REF] Kamble | Analysis of the driving and dependence power of barriers to adopt industry 4.0 in Indian manufacturing industry[END_REF].

Previous studies agree that investigations of challenges faced by firms in digitalization have not been sufficiently explored [START_REF] Frank | Industry 4.0 technologies: Implementation patterns in manufacturing companies[END_REF]Horváth and Szabó, 2019) and that more research is required to achieve a general consensus.

On the one hand, most research states, based on the Resource-Based View (RBV - [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF] that firm size is one of the determining factors in SC digitalization (Yang et al., 2021) along with SC maturity (Zouari et al., 2021) which requires competences [START_REF] Colli | A maturity assessment approach for conceiving context-specific roadmaps in the Industry 4.0 era[END_REF], an organizational structure (Schuh et al., 2017) and the capability to capture value [START_REF] Schumacher | A maturity model for assessing Industry 4.0 readiness and maturity of manufacturing enterprises[END_REF]. Our literature review reveals the heterogeneity of the corpus concerning the link between firm size and digital transformation. The adoption of digital tools may require significant investment and expertise [START_REF] Williams | The unified theory of acceptance and use of technology (UTAUT): a literature review[END_REF][START_REF] Pan | Logistics IS resources, organizational factors, and operational performance: An investigation into domestic logistics firms in China[END_REF], and smaller firms have limited resources available for such investments [START_REF] Elia | Resources and digital export: An RBV perspective on the role of digital technologies and capabilities in cross-border ecommerce[END_REF].

However, some research points out that SMEs are more flexible as the decision-making and information are decentralized, which facilitates the digital transformation (Müller et al., 2018b;Gupta et al., 2020).

On the other hand, [START_REF] Lin | Strategic response to Industry 4.0: an empirical investigation on the Chinese automotive industry[END_REF] show that firm size and nature1 do not have any significant impact on firms' digitalization. This means that, regardless of their size, firms highlight their difficulties in digitalizing their SCs (Deloitte, 2017). Thus, firm size is a factor that has an unclear and ambiguous position in the SC digitalization process.

The contribution of this study is to shed new light on the role of firm size on the SC digitalization. Indeed, if the firm size is a usual control variable in this line of research, the focus is never on this variable and the way it directly impacts the SC digitalization and its challenges.

Therefore, to bridge this gap, we attempt to answer the following research question: Is there any relationship between the size of the firm and (i) its SC digital tools adoption level and (ii) the challenges to SC digitalization? Addressing this gap is an essential contribution for research to consider SC digitalization more realistically: although SC digitalization may bring advantages for some firms, most of them, whatever their size, struggle with it and many SC professionals do not know where to begin in the process [START_REF] Whysall | The new talent management challenges of Industry 4.0[END_REF].

From a theoretical standpoint, we mobilize RBV related to the availability of resources for Supply Chain Management (SCM) [START_REF] Elia | Resources and digital export: An RBV perspective on the role of digital technologies and capabilities in cross-border ecommerce[END_REF][START_REF] El Baz | Can supply chain risk management practices mitigate the disruption impacts on supply chains' resilience and robustness? Evidence from an empirical survey in a COVID-19 outbreak era[END_REF]. Indeed, RBV is a relevant framework because research shows that large firms have greater access to resources [START_REF] Asamoah | Interorganizational systems use and supply chain performance: Mediating role of supply chain management capabilities[END_REF].

From a practical standpoint, our study offers several managerial contributions by better embracing the reality of firms trying to digitalize their SC and theoretical contributions by exploring the aspects of SC digitalization that are impacted or not by firm size, which makes it possible to clear up ideas about firm size and SC digitalization that are sometimes preconceived for several academics and practitioners.

To reach our research goal, 311 surveys from SCM professionals were statistically analyzed.

Three steps were necessary: (i) defining each of the challenges faced by firms in the digitalization context (as per [START_REF] Türkeș | Drivers and barriers in using industry 4.0: a perspective of SMEs in Romania[END_REF][START_REF] Raj | Barriers to the adoption of industry 4.0 technologies in the manufacturing sector: An inter-country comparative perspective[END_REF]; (ii) measuring the SC digital tools adoption level; (iii) evaluating the impact of firm size from an RBV perspective, on those SC digitalization variables [START_REF] Li | The impact of digital technologies on economic and environmental performance in the context of industry 4.0: A moderated mediation model[END_REF].

The paper is structured as follows: after having laid the theoretical foundations of our analysis, we review the relevant literature on SC digitalization and SC digital tools adoption and describe the challenges faced by firms when digitalizing their SCs. We then discuss the methodological approach and subsequently present and discuss the results. Finally, our conclusion provides the contributions, limitations, and future research perspectives.

Theoretical background and hypotheses development

Firm size and supply chain digitalization from RBV perspective

The need for SC digitalization in our current time of recurrent crises can be analyzed through the lenses of the RBV [START_REF] Wamba | Industry 4.0 and the supply chain digitalisation: a blockchain diffusion perspective[END_REF]. RBV is a well-established theoretical approach that highlights that firms may achieve long-term competitive advantages if they possess valuable, inimitable, and non-substitutable resources [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF]. According to Barney (1991, p. 114), "computers or other types of machines, are part of the physical technology of a firm and usually can be purchased across markets." Thus, the links between RBV and digitalization have been highlighted since the origin of this theoretical framework.

The contribution of technologies to the development of competitive advantages is widely emphasized in academic research [START_REF] Wamba | Industry 4.0 and the supply chain digitalisation: a blockchain diffusion perspective[END_REF][START_REF] Ivanov | Researchers' Perspectives on Industry 4.0: Multi-Disciplinary Analysis and Opportunities for Operations Management[END_REF] and RBV is a popular theoretical framework for studying this [START_REF] Seyedghorban | Supply chain digitalization: past, present and future[END_REF]. At the same time, studies highlight difficulties for small firms in identifying, acquiring, and developing their resources, including those related to digitalization [START_REF] Elia | Resources and digital export: An RBV perspective on the role of digital technologies and capabilities in cross-border ecommerce[END_REF]. Indeed, firm size reflects the amount of available resources that can be leveraged to build a competitive advantage [START_REF] Florez-Lopez | Resource-based view and SMEs performance exporting through foreign intermediaries: The mediating effect of management controls[END_REF]; the smaller the firm, the fewer resources it has. However, the few research studies that mobilize the 'firm size' variable in the context of SC digitalization find results that are sometimes contradictory (e.g. [START_REF] Lin | Strategic response to Industry 4.0: an empirical investigation on the Chinese automotive industry[END_REF] stating that firm size does not have any significant impact on digitalization whereas Yang et al. (2021) show a positive impact), suggesting that resources to achieve this digitalization are certainly lacking in smaller firms but may also be lacking in larger firms. In the next section, we develop these aspects more thoroughly.

Supply chain digitalization

As a new industrial revolution, digitalization is one of the most popular topics among international institutions and professionals; thus, it is a burgeoning research area [START_REF] Raj | Barriers to the adoption of industry 4.0 technologies in the manufacturing sector: An inter-country comparative perspective[END_REF]. Due to this revolution, numerous business models and organizations are changing and will continue to change profoundly [START_REF] Büyüközkan | Digital Supply Chain: Literature review and a proposed framework for future research[END_REF]. Lately, the rise in the use of interactive and mobile communication tools has accelerated this revolution [START_REF] Schniederjans | Supply chain digitisation trends: An integration of knowledge management[END_REF][START_REF] Ivanov | Digital supply chain management and technology to enhance resilience by building and using end-to-end visibility during the COVID-19 pandemic[END_REF]. In this context, which frequently highlights the expected benefits of digitalization, other perspectives are being examined. For instance, a recent literature review [START_REF] Ain | Two decades of research on business intelligence system adoption, utilization and success-A systematic literature review[END_REF] indicates that researchers and practitioners are also interested in better understanding the challenges faced when implementing digital tools. [START_REF] Ivanov | Digital supply chain management and technology to enhance resilience by building and using end-to-end visibility during the COVID-19 pandemic[END_REF] mentions the important challenges to create end-to-end visibility in the SC thanks to digital tools.

Examining SC digitalization specifically, SCM is considered to be a very suitable application domain for digital tools [START_REF] Park | Visual analytics for supply network management: System design and evaluation[END_REF]. Indeed, such tools enhance the decision-making processes of SC managers. In this context, Büyüközkan and Göçer (2018, p.165) define digital SC as "an intelligent best-fit technological system that is based on the capability of massive data disposal and excellent cooperation and communication for digital hardware, software, and networks to support and synchronize interaction between organizations by making services more valuable, accessible and affordable with consistent, agile and effective outcomes."

Digital tools for Supply Chain Management and their adoption level

Digital tools are needed to enhance the decision-making processes of SC managers [START_REF] Park | Visual analytics for supply network management: System design and evaluation[END_REF] and disrupt traditional SCs. These new technologies are directed to distributed systems in order to improve physical and communication systems. They work with different methods like auto-optimization, simulation, intelligent worker support, self-diagnosis, machine perception and self-configuring in order to achieve multiple goals. For example, tools such as the Internet of Things (IoT) play a key role in the design, operations, and performance of global SCs (Gunasekaran et al., 2016). Zouari et al. (2021) study 15 digital tools which are the most useful in the field of SCM and list their definitions. Based on this research, in Table I we consider the same tools and explain their main purpose in SCM.

---Table I ---Digital tools are essential to the success of global SC networks (Gunasekaran et al., 2016) and are desired by most firms, even smaller ones, because they recognize the benefits over the costs and organizational challenges [START_REF] Buer | The digitalization of manufacturing: investigating the impact of production environment and company size[END_REF]. Future SC will converge people, businesses, and things in a digital value network thanks to the incorporation of fast-emerging digital tools. In the context of SCM, the adoption of digital tools seeks to achieve not only coordination inside the firm but also inter-organizational coordination by means of electronic links between information systems, enabling automated and digitalized processes which involve different partners from suppliers to customers [START_REF] Büyüközkan | Digital Supply Chain: Literature review and a proposed framework for future research[END_REF].

Several models for digital tools adoption exist and these may help to understand the various adoption stages (e.g., Hameed et al., 2012), from the pre-adoption phase and its antecedents to the implementation phase, including users' practices and habits, and then to the post-adoption phase. However, these models, which focus on the adoption phases of digital tools, do not provide information about their adoption level. Nevertheless, this is of key importance: the emergence of Industry 4.0 has brought an important number of challenges and opportunities for organizations and with these, the need to evaluate the digitalization process.

Challenges to Supply Chain digital tools adoption

A number of scholars underline the need for further investigation of the challenges to achieving digital adoption [START_REF] Raj | Barriers to the adoption of industry 4.0 technologies in the manufacturing sector: An inter-country comparative perspective[END_REF]. Indeed, SC digitalization is still in its nascent stage; the transformation into an effective digital SC requires specific capabilities and firms need to develop a deep understanding of their current situation. Numerous studies agree that the lack of skilled workflow and training to fit the change is one of the major challenges in digital tools implementation [START_REF] Büyüközkan | Digital Supply Chain: Literature review and a proposed framework for future research[END_REF][START_REF] Raj | Barriers to the adoption of industry 4.0 technologies in the manufacturing sector: An inter-country comparative perspective[END_REF]. Türkes et al.

(2019) highlight the organizational resistance from employees and middle management levels, and also the lack of expertise.

It is also important to highlight that the integration of systems, tools, and methods requires the development of a flexible interface for the synchronization of different languages, technologies, and such methods can lead to significant challenges in SC digital tools adoption [START_REF] Büyüközkan | Digital Supply Chain: Literature review and a proposed framework for future research[END_REF]Müller, et al., 2018b).

In Table II, we highlight the various challenges identified from the literature ---Table II ---

The ambiguous influence of firm size on digitalization

Prior research on digital tools adoption reveals that "firm size" is a potential determinant of the adoption process [START_REF] Frank | Industry 4.0 technologies: Implementation patterns in manufacturing companies[END_REF]Gupta et al., 2020;[START_REF] Li | The impact of digital technologies on economic and environmental performance in the context of industry 4.0: A moderated mediation model[END_REF] because firm size influences the quantity of resources available to succeed with digitalization, e.g., technological and financial resources, organizational skills and knowledge and expertise (Ramon-Jeromino et al., 2019;[START_REF] Pan | Logistics IS resources, organizational factors, and operational performance: An investigation into domestic logistics firms in China[END_REF][START_REF] Buer | The digitalization of manufacturing: investigating the impact of production environment and company size[END_REF][START_REF] Elia | Resources and digital export: An RBV perspective on the role of digital technologies and capabilities in cross-border ecommerce[END_REF]. The size may be measured by the number of employees, turnover, total assets, or even market shares.

The smallest firms lag behind larger ones regarding digital transformation [START_REF] Mittal | A critical review of smart manufacturing & Industry 4.0 maturity models: Implications for small and medium-sized enterprises (SMEs)[END_REF] and are ill-prepared for the new technologies' changes [START_REF] Basl | Pilot study of readiness of Czech companies to implement the principles of Industry 4.0[END_REF]. Indeed, large firms have more resources to invest in digital tools and are more capable of bearing the high risk associated with this investment. Furthermore, small and medium-sized firms tend to underestimate the time, skills, and effort required to adopt digital technologies [START_REF] Basl | Pilot study of readiness of Czech companies to implement the principles of Industry 4.0[END_REF]. Some researchers note that the shortage of financial resources could be a significant obstacle to digitalization for SMEs (Müller et al., 2018b;[START_REF] Basl | Pilot study of readiness of Czech companies to implement the principles of Industry 4.0[END_REF]. [START_REF] Lee | A Big Data Analytics Platform for Smart Factories in Small and Medium-sized Manufacturing Enterprises: An Empirical Case Study of a Die Casting Factory[END_REF] highlight that SMEs lack financial resources when implementing Radio Frequency Identification (RFID) as well as for using data from this technology. The authors also remark that a lack of human competency is an obstacle when attempting to implement data mining technologies. In addition, some governments provide financial support for firms wanting to digitalize their processes, but many small firms do not meet the size requirements to benefit from those programs (Gessner and Snodgrass, 2015).

Nevertheless, it should be stressed that SMEs possess certain strengths that are harder for larger firms to reproduce. They usually innovate much faster than established large firms since they have flexible organizational structures [START_REF] Bouncken | Knowledgeand innovation-based business models for future growth: Digitalized business models and portfolio considerations[END_REF]. Furthermore, SMEs are recognized as being more open to IT than larger firms and are ready to form alliances when size does not permit the use of technology advancements [START_REF] Narula | R&D Collaboration by SMEs: new opportunities and limitations in the face of globalization[END_REF]. In addition, depending on the industry, SMEs do not always lag behind: for example, in export activities that require important investment in digital tools for SCM, they show no less adoption of digital tools despite their limited resources [START_REF] Elia | Resources and digital export: An RBV perspective on the role of digital technologies and capabilities in cross-border ecommerce[END_REF]. Another example comes from [START_REF] Buer | The digitalization of manufacturing: investigating the impact of production environment and company size[END_REF], who show that "firm size" does not always have an impact on SC digitalization. More specifically, in this study, on the one hand, firm size influences the digitalization of warehouses, but on the other hand, the size has no impact on the adoption of digital tools that support the SC integration. Buer et al. (2020, p. 635) suggest that this result could mean that "such solutions [for SC integration] have started to become mainstream."

Finally, in their study on the strategic response to digitalization, [START_REF] Lin | Strategic response to Industry 4.0: an empirical investigation on the Chinese automotive industry[END_REF] show that firm size and nature do not have any significant impact on the digitalization of firms, which may sound like a surprising result.

Therefore, we can conclude that both small and large firms have different opportunities (Horváth and Szabó, 2019) and the success or failure of their SC digitalization is determined by several factors. Overall, the relationships between firm size and digital adoption have been explored in relatively little detail, and the results appear contradictory, because firm size may have various effects on the different aspects of SC digitalization. Furthermore, few authors have made an empirical examination of the challenges to digital transformation and the effect of firm size remains unclear.

To answer the research questions, we propose two hypotheses: H1: There is a significant difference between groups of firms of different sizes regarding the adoption of supply chain digital tools.

H2: There is a significant difference between groups of firms of different sizes regarding the challenges to supply chain digitalization.

Research objectives and methodology

Research objectives

We aim to better understand and analyze which digital tools firms adopt for their supply chain and what the level of adoption is for each of those tools, along with what the various challenges are that firms must face in their digitalization program and, more importantly, what the impact is of firm size on SC digital tools adoption and challenges to SC digitalization.

Overall, these three steps contribute to answering the research question.

Data collection and sampling method

We carried out a quantitative study to measure the phenomenon of SC digitalization through digital tools adoption, challenges to this digitalization, and the impact of firm size on those two variables. For this purpose, we created an online survey after reviewing the relevant academic literature. We chose Google Forms as a platform as it allowed us to retrieve all the answers on a spreadsheet in chronological order, together with descriptive statistics of the collected data. Prior to any diffusion of the survey, we followed a validity process by pretesting it on 10 people: two senior SC Managers and eight researchers. This pretest led us to eliminate one item: Advanced Human-Technology Interface (AHTI). Indeed, several respondents indicated that they were unsure of identifying which interfaces we were referring to.

The survey is divided into three parts: digital tools adoption (14 tools after the exclusion of AHTI -see Table I), challenges to digitalization (9 items -see Table II), and some general questions such as firm size. All items and tools were chosen following the literature review. In this survey, we are interested in managers' perceptions and representations concerning SC digitalization within their company.

We collected the data in November 2020. To collect 311 complete and valid questionnaires, 2320 personal emails were sent to various qualified SC managers in different firms in Europe.

No technicians were included in the sample. The set of respondents was selected according to the "convenience sample" principle which is frequently used in SCM research (De Beuckelaer and Wagner, 2021) despite the potential lack of generalizability. Indeed, all the professionals mobilized are linked to the education programs provided by the university institutions to which the members of our research team belong. The advantages of using a convenience sample is the possibility: (i) to control the quality of the profiles of the potential respondents and (ii) to ensure there is only one respondent per company. In addition, this type of survey for collecting data is relevant when there are small firms in the sample [START_REF] Flynn | Survey research design in supply chain management: the need for evolution in our expectations[END_REF].

The response rate is 13.41%, which is acceptable [START_REF] Dillman | Mail and Internet Surveys: The Tailored Design Method[END_REF]. We analyzed the dataset using IBM SPSS statistics 27.

Response and common method bias

To achieve a high response rate and avoid non-response bias, we included in the emails the personalized assurance that the results would be sent as soon as they are available [START_REF] Frohlich | echniques for improving response rates in OM survey research[END_REF]. Furthermore, we assessed non-response bias using the protocol proposed by [START_REF] Rogelberg | Introduction: Understanding and dealing with organizational survey nonresponse[END_REF]. We evaluated and compared the early 25% of respondents with the last 25% ones. This technique enabled us to detect if any difference existed between their responses. A t-test comparison between these two groups regarding two descriptive variables-the number of employees and company turnover-indicated no significant difference in the responses of early and late participants (p<0.05). These results indicate that common method bias is not an issue for this study.

Empirical results

Brief description of our sample

Regarding firm size, 46.2% of the respondents work in firms of more than 5000 employees with an international dimension. We should note that, for a study about the impact of firm size, many of the respondents are from large firms. This is because we sought responses from SC Vice Presidents, Directors and Managers, profiles mostly found in large firms, as the SCM function is still under the process of development in small firms [START_REF] Kilpi | The effect of external supply knowledge acquisition, development activities and organizational status on the supply performance of SMEs[END_REF] and with a low level of SC maturity (Zouari et al., 2021). Nevertheless, our study also includes respondents working for SMEs in their local market (20% of respondents).

More than half of the sample (67.2%) work in the manufacturing industry, the rest mainly working in retail or services (e.g., transportation). In addition, 69.5% of respondents have at least 9-years' experience in an SCM-related field. The respondents work mainly at the managerial level (83.28%), with positions such as Vice-President, Directors, and Managers.

Given the extensive corporate experience in SCM, our sample has abundant knowledge of SCM and is in a position to take decisions regarding SC digitalization projects.

SC digital tools adoption

In this section, we first depict the means and standard deviation for each item related to digital tools adoption in an SCM context in order to better embrace the current reality of firms; subsequently, we investigate the impact of firm size on adoption.

Digital tools adoption in Supply Chain Management

This study measures the adoption of 14 digital tools used in the SCM field. Recent studies have frequently focused on several of those digital tools in order to perceive what advantages they could bring to SCM. The overall impression is that digital tools are widely adopted through SCs.

Very recently, some researchers have offered scales to evaluate the adoption degree of digital tools in SCM [START_REF] Li | The impact of digital technologies on economic and environmental performance in the context of industry 4.0: A moderated mediation model[END_REF]Zouari et al., 2021). More precisely, [START_REF] Li | The impact of digital technologies on economic and environmental performance in the context of industry 4.0: A moderated mediation model[END_REF] study the implementation of four SC digital tools and evaluate the extent to which firms have implemented digital technologies in their operation by using a 7-point Likert scale. However, they do not explicitly state the meaning of each of the 7 points. In contrast, Zouari et al.

(2021) propose a measurement scale for digital tools adoption in the field of SCM which includes five levels and the definition of each (see Table III). We, therefore, adopt this scale for our study. The question we asked is "Please indicate your level of adoption of the following digital technologies/tools to help improve your supply chain management." For each tool, respondents scored the adoption level within their firm: (1) not started, (2) basic,

(3) developing, (4) intermediate, and (5) advanced.

---Table III ---Table IV shows the adoption levels and ranks the digital tools, from the most adopted to the lowest.

---Table IV ---Unsurprisingly, collaborative tools are the most adopted (3.61): they are part of the oldest and best-known tools developed in SCM. Nevertheless, on average participants score them only between (3) and ( 4), when we might expect a stronger adoption. All the other digital tools are scored lower than (3) and half are even lower than (2). For the less adopted tools, the standard deviation is also low (around 1), which reflects the strong unanimity of the responses. A surprising answer emerges concerning the low adoption level of artificial intelligence (AI) and machine learning (ML): 38.9% affirm the development stage of AI is at level (1), 34.1% report the same about ML at a time when academic literature on this topic is strongly increasing [START_REF] Sharma | A systematic literature review on machine learning applications for sustainable agriculture supply chain performance[END_REF]. Finally, it appears that, overall, the adoption level of almost every digital tool is weak in SCM.

Relationship between firm size and SC digital tools adoption

"Firm size" is frequently considered a variable with an influence over topics related to digitalization. In this section, we explore in detail the potential link between digital tools adoption and firm size. We separated respondents into three groups depending on their firms' employee numbers. We used the firm size categories defined by the National Institute of Statistics and Economic Studies (INSEE), which collects, analyses, and disseminates information on the French economy and society and is an internationally recognized Institute [START_REF] Stekelorum | Cooperation with international NGOs and supplier assessment: Investigating the multiple mediating role of CSR activities in SMEs[END_REF]. According to this institute, firms that employ fewer than 249 people are defined as small enterprises, those with 250-4999 employees as medium-sized enterprises, and those with 5000 or more employees as large enterprises. Firm size is a categorical variable, and respondents answered the question "What is the approximate number of employees in your company?".

In this study, Group 1 (G1) is composed of the largest firms (more than 5000 employees; 46.2% of the total sample), Group 2 (G2) is the average range (250 to 5000; 33.8%) and Group 3 (G3) has the smallest firms (less than 250; 20%).

To test the link between firm size and digital tools adoption, we first ran a factorial analysis (EFA) with all 14 items using maximum-likelihood extraction with varimax rotation [START_REF] Corner | Choosing the right type of rotation in PCA and EFA[END_REF] in order to avoid the Type I errors implied by running a large series of tests [START_REF] Field | Discovering statistics using IBM SPSS statistics[END_REF]. Consequently, all items were retained with 2 dimensions. Every dimension contains elements homogeneously grouped (see Table V): construct (1) includes operational digital tools and construct (2) modern decision support digital tools.

The EFA explained 55.88% of the variance for a Kaiser-Meyer-Olkin (KMO) of .930, which is excellent according to Hutcheson and Sofroniou (1999). We then assessed the reliability of the scale by Cronbach's alpha; Cronbach's α values were above the lower limit of .60 (Hair et al., 2017).

---Table V ---

We then ran Kruskal-Wallis tests as not all our data follow a normal distribution [START_REF] Landau | A handbook of statistical analyses using SPSS[END_REF]), which we established after running the Shapiro-Wilk test [START_REF] Shapiro | An analysis of variance test for normality (complete samples)[END_REF]. The Kruskal-Wallis test is not sensitive to the normal distribution [START_REF] Landau | A handbook of statistical analyses using SPSS[END_REF]; this test is a widely used non-parametric technique for testing different samples from the same population [START_REF] Daniel | Applied nonparametric statistics[END_REF]. It does not show how the groups differ, it only tells us that they are different in some manner [START_REF] Chan | Learning and understanding the Kruskal-Wallis one-way analysis-of-variance-by-ranks test for differences among three or more independent groups[END_REF]. The Kruskal-Wallis tests results are presented in Table VI. They show that firm size influences both constructs of SC digital tools. Thus, H1 is validated.

---Table VI ---

As the Kruskal-Wallis tests show a statistically significant difference, we perform a post hoc analysis (Mann-Whitney U tests) to check between which groups there are actually statistically significant differences. At this stage, the Bonferroni correction should be calculated [START_REF] Field | Discovering statistics using IBM SPSS statistics[END_REF] to determine the new alpha level. As there are 3 groups in our study, the calculation is as follows [START_REF] Field | Discovering statistics using IBM SPSS statistics[END_REF]: 0.05/(3*(3-1)/2)=0.017. Table VII provides the results and shows that, regarding the first set of SC digital tools, those that are more operations-oriented, differences between groups exist between G1 vs G2 and G1 vs G3. This means that the largest firms are significantly more advanced in their operational digital tool adoption than the medium and small firms. Examining the means of the adoption levels enables us to understand better the differences between groups. Therefore, the operational digital tool adoption means score for G1 is 2.38, G2=2.02, and G3=1.69. With regards to SC digital tools which are decision support-oriented, here differences between groups exist between G1 vs G3 and G2 vs G3. This means that the smallest firms are significantly less advanced in their decision support digital tool adoption than the large and medium firms (decision-support digital tool adoption mean score for G1=1.76, G2=1.53, G3=1.22).

---Table VII ---

Challenges to digitalization

In this section, we first detail the means and standard deviation for each item related to challenges to SC digitalization in order to better understand them, and subsequently investigate the impact of firm size on these.

Challenges to Supply Chain Digitalization

We also explore the challenges faced by the respondents and their firms in improving the digitalization of their SCs. Respondents answered the question "What factors are limiting your organization's ability to capture value from existing digital opportunities in the supply chain?" for each item (each of the 9 challenges). They had to score on a Likert scale between (1) Totally disagree and (5) Total agree. Table VIII shows the scores for each challenge.

---Table VIII ---Focusing on the TOP 3 challenges to SC digitalization, it appears that the first is mainly linked to technology and human-technology interplay whereas the two others are more human-process interplay-related.

Relationship between firm size and challenges to supply chain digitalization

Following the same procedure as the previous one (see section 5.2.2.), to test the link between firm size and the challenges to supply chain digitalization, we first ran the factorial analysis (EFA) with all 9 items using maximum-likelihood extraction with varimax rotation. The results show that 8 items were retained within the same dimension (see Table IX) andthat C9 shows a very high loading (above .9, more precisely .925), which justifies its exclusion [START_REF] Field | Discovering statistics using IBM SPSS statistics[END_REF]. The EFA explained 53.44% of the variance for a KMO of .901. The reliability analysis shows Cronbach's α value of .871.

---Table IX ---Then, considering that the Shapiro-Wilk test showed again that the data does not follow a normal distribution, we ran a Kruskal Wallis test. The results are presented in Table X and the correlation between firm size and the challenges to SC digitalization were not statistically significant, which indicates the hypothesis H2 was not supported. Thus, H2 is rejected and there was no need to further investigate by running Mann-Whitney tests. This implies that, regardless of firm size, there is no significant difference in the manner firms experience the challenges of SC digitalization.

---Table X ---

Discussion

In this article, we aim to better understand the relationships between firm size and SC digitalization by examining the adoption level of SC digital tools and the challenges linked to SC digitalization. The first step in answering the research question concerned the current adoption levels of SC digital tools; results are shown in Table IV. The findings regarding AI and blockchain technology require further discussion, as the descriptive statistics show that these are among the least-adopted SC digital tools. However, [START_REF] Wang | Understanding blockchain technology for future supply chains: a systematic literature review and research agenda[END_REF] conducted a systematic literature review on blockchain technology applications in SCs and offer a research agenda indicating an ever-growing interest from researchers in this tool. Similarly, [START_REF] Toorajipour | Artificial intelligence in supply chain management: A systematic literature review[END_REF] conducted a systematic review of the literature on AI in SCM and show the increasing number of studies on those tools, which are considered the most promising for SCM in the future. Therefore, there appears to be a dissonance with firms' reality and it is necessary to question the validity of these studies if the research does not accompany the experience of firms [START_REF] Stentoft | Balancing theoretical and practical relevance in supply chain management research[END_REF] in the adoption of these digital tools. Beyond operational research highlighting the possible advantages of AI or blockchain technology (e.g., [START_REF] Liu | A blockchain-based framework of cross-border e-commerce supply chain[END_REF] for SCs, qualitative research to better explore the obstacles and remove them in an action research dynamic is also desirable.

Another interesting result related to the adoption level of SC digital tools is based on the factorial analysis of the 14 SC digital tools considered in this study. The statistical analysis shows that two categories emerge, based on the level of adoption: the first category contains digital operational tools (i.e., SDV, ROBOT, SS, COLLAB, MDW, LDT, SMT, and CLOUD) and the second the modern decision support digital tools (i.e., AI, ML, BDA, BLOCK, VR, and IoT). An interesting outcome of this research is the discovery of a new form of categorization of tools, based on the level of adoption, at a time when some researchers aim to propose such typologies (e.g., Ivanov et al., 2019). Examining the two categories, it is clear that in terms of operational digital tools, the largest firms (G1) are significantly more advanced than the others (G2 and G3). However, for the so-called decision-oriented modern digital tools, G1 and G2 show no difference between them, whereas they both differ significantly from G3. However, it is not possible to conclude that G1 and G2 firms are significantly more "advanced" in their adoption than G3, because referring to them as advanced is not realistic. Whereas G1 and G2 firms show a more advanced adoption of these digital tools than those in G3, the average adoption levels remain very low, and lower than for the category of operational tools. This is an important finding at a time when consultants and researchers are focusing a great deal on tools such as AI, Big Data Analytics, or blockchain: on the ground, the level of adoption remains very low [START_REF] Brinch | Practitioners understanding of big data and its applications in supply chain management[END_REF] and the need to support firms, regardless of their size, is very important.

In respect of the challenges to SC digitalization, which is the second step in our research, the findings regarding "collaboration" (C2) for the whole sample are particularly interesting.

Collaboration is always a paradoxical theme for SCM research. Whereas so-called "collaborative" tools are the most developed (TOP 1), the statistical results indicate that "lack of collaboration with SC partners" ranked 6th among the 9 challenges to digitalization. This finding is to be considered in the light of pre-existing research. Collaboration in the SCM field is still a common topic. Over the last two decades, thousands of articles have been published, reflecting the considerable interest in a concept that is difficult to assess empirically [START_REF] Fawcett | Why supply chain collaboration fails: the socio-structural view of resistance to relational strategies[END_REF]. Therefore, research on SC collaboration is still being conducted and frequently with a focus on technology (e.g., [START_REF] Chi | Collaboration in demand-driven supply chain: Based on a perspective of governance and IT-business strategic alignment[END_REF]. This finding shows the difficulty of simultaneously relying on SC partners when a firm lacks competencies and resources for SC digitalization and of collaborating with the above-mentioned partners.

Finally, with regard to "firm size," research has widely highlighted the influence of this variable on digitalization issues (e.g., [START_REF] Frank | Industry 4.0 technologies: Implementation patterns in manufacturing companies[END_REF]Gupta et al., 2020;[START_REF] Li | The impact of digital technologies on economic and environmental performance in the context of industry 4.0: A moderated mediation model[END_REF], whereas fewer studies have shown that smaller firms also have advantages in the digitalization race [START_REF] Narula | R&D Collaboration by SMEs: new opportunities and limitations in the face of globalization[END_REF][START_REF] Bouncken | Knowledgeand innovation-based business models for future growth: Digitalized business models and portfolio considerations[END_REF]. Our research finds results that contradict the mainstream concept that firm size is a key variable to consider when addressing SC digitalization. The existence, on the one hand, of a dominant research stream that strongly suggests the significant influence of firm size on SC digitalization in general and, on the other, scientific publications valuing the innovativeness in this field among SMEs or managerial publications showing the difficulties for all kinds of firms faced with digitalization (Deloitte, 2017), implies a great deal of ambiguity concerning the real influence of firm size.

Our results highlight two main findings about the relationships between firm size and the two variables: the adoption level of digital tools and challenges to SC digitalization. Our results show that firm size has a significant correlation with the level of adoption of digital tools. On the other hand, our findings confirm that there is no significant correlation between firm size and the challenges to SC digitalization, as indicated by the results of the Kruskal-Wallis test.

Therefore, almost all firms, whatever their size, face identical challenges when trying to adopt SC digital tools. This means that the findings do not fully corroborate suggestions in the RBV regarding the importance of resources in succeeding with SC digitalization [START_REF] Florez-Lopez | Resource-based view and SMEs performance exporting through foreign intermediaries: The mediating effect of management controls[END_REF][START_REF] Elia | Resources and digital export: An RBV perspective on the role of digital technologies and capabilities in cross-border ecommerce[END_REF]; even the largest firms face challenges and have as much difficulty as smaller ones in meeting them. This result suggests that all kinds of firms currently need to acquire new resources, certainly in the form of knowledge and skills, seek to gain digital maturity, and succeed in their SC digitalization projects beyond the simple adoption of digital tools (Zouari et al., 2021). Of course, it is natural to anticipate that larger firms will have an easier time acquiring these necessary resources first, just as they were the first to be capable of adopting SC digital tools.

Conclusion

The findings discussed above provide several insights for research and practice and open various research avenues.

Theoretical implications and contributions

In this study, in order to contribute to the line of research on SC digitalization, we aimed to understand and analyze the relationships between firm size and SC digitalization, more precisely on the adoption level of SC digital tools and the challenges linked to SC digitalization. This enabled us to develop a realistic picture of the adoption of SC digital tools and the main challenges faced by firms when they consider digitalizing their SC. More importantly, we contribute to the scientific literature by identifying in which respects firm size matters, in order to address the ambiguity stemming from earlier research studies. To do so, we referred to the RBV as a theoretical lens, which enabled us to understand how firm size and related resources have an ambivalent role in SC digitalization, which we develop hereafter for a deeper understanding.

The findings show, first, that SC digital tools can be divided into two categories (operational and modern decision-support digital tools) and that the adoption level of SC digital tools is mostly at the "basic" stage for firms. Notably, AI and blockchain are very commonly studied in SCM research [START_REF] Wang | Understanding blockchain technology for future supply chains: a systematic literature review and research agenda[END_REF][START_REF] Toorajipour | Artificial intelligence in supply chain management: A systematic literature review[END_REF], although this does not reflect the reality of all firms. This suggests that researchers should consider how to better address the different realities of firms regarding digitalization, and, in particular, those of SMEs, which constitute a large part of the global business environment 2 . Second, the findings indicate that almost all the challenges to SC digitalization were higher than average for the respondents. Having established this, we then provided multivariate analyses to assess the link between firm size and the aforementioned variables. The key finding from this is as follows: firm size does not have a link to every aspect of SC digitalization, which makes it possible to clarify and provide nuance for some of the previous research statements. If different groups of firms categorized by size show significant differences in their adoption level of SC digital tools, it appears that such variation does not exist for the challenges to SC digitalization. These contributions provide a better understanding that firm size does not always have the same types of relationships with different aspects of SC digitalization. This is the main theoretical contribution of this study. Therefore, SC managers and researchers may consider firm size, as a key aspect of the RBV, and which has a direct effect on the resources available, as having an ambivalent role in SC digitalization.

Practical implications and contributions

As a first managerial contribution, practitioners should find out more about the opportunities offered by advanced technologies. New digital tools appear every day and companies have an interest in following this progress and taking advantage of it. We believe that digital tools boost firms' creativity and help them capture new market opportunities. Regardless of firm size, digital transformation provides more available data and enhances the efficiency of the process, as well as communication. Barriers to digitalization are the same for all kinds of firms and may significantly inhibit the level of adoption of SC digital tools. Therefore, managers should understand and mitigate these challenges for successful adoption, which is not possible without the development of a digital culture, greater digital maturity, and more expertise. Given the pressure placed on SC managers by the COVID-19 outbreak, the adoption rate of digital tools is accelerating [START_REF] Modgil | Has Covid-19 accelerated opportunities for digital entrepreneurship? An Indian perspective[END_REF]; without enhanced skills, training, and support with working in a digital SC, employees are unlikely to adopt new technologies. As a final contribution, we also urge practitioners to explore further the methods by which they can collaborate with their SC partners to improve their digitalization and together remove the barriers to it.

Limitations and further research directions

All research contains limitations that are often opportunities for future research. The limitations of our paper include methodological concerns. First, the sample was composed of 311 qualified respondents from firms with differing characteristics from several industries, but exclusively from European countries. Second, we did not test a conceptual model requiring the use of more advanced statistical methods. Nevertheless, this was not necessary to examine both the reality of the level of adoption of SC digital tools and the challenges to SC digitalization faced by firms.

The type of statistical analysis conducted in this study has been proposed due to the descriptive nature of this study. With regard to further research, we are confident that future hypothetical models combining several variables from the SC digitalization research field through PLS-SEM techniques will consider the potential influence of "firm size" in a more precise manner. Future research could consider other key factors that may influence the implementation of digital tools in the SC, such as the level of engagement of the organization in SC digitalization and its digital maturity, which will make it possible to ascertain whether the companies involved have better control of the barriers encountered.

In addition, in an SC context, it would also be interesting for future research to examine users' perspective on SC tools adoption and acceptance by mobilizing the technology acceptance model [START_REF] Davis | User acceptance of computer technology: a comparison of two theoretical models[END_REF] or the unified theory of acceptance and use of technology [START_REF] Venkatesh | User acceptance of information technology: Toward a unified view[END_REF].

Despite the limitations presented here, the authors believe that this study provides valuable insights and reveals implications that serve both researchers and practitioners in better understanding relevant issues linked to SC digitalization.

  Yang, L., Huo, B., Tian, M., & Han, Z. (2021) 'The impact of digitalization and interorganizational technological activities on supplier opportunism: the moderating role of relational ties', International Journal of Operations & Production Management, Vol. 41 No. 7, pp.1085-1118. Zouari, D., Ruel, S., & Viale, L. (2021) 'Does digitalising the supply chain contribute to its resilience?', International Journal of Physical Distribution & Logistics Management, Vol. 51 No. 2, pp.149-180.

Table I .

 I as on demand access to inventory information, as well as massive scalability in service, payment and privatization. Companies can use it to share real-time overviews of inventory and sales information resulting in closer integration between channels and more efficient SC and Digital tools and their purpose within SCM

			customer analytics.
	Digital tools		N	Mean	Standard	RANK
					deviation
	COLLAB			311	3.61	1.252	1
	Blockchain MDW	BCT	BCT contains a single record of the data which is 311 2.85 1.369	Hughes et al., (2019) 2
	technology		stored by every participant in the SC, so it enforces
	CLOUD		transparency. BCT enables cutting timeframes in SC 311 2.75 1.415	3
			financing.
	ROBOT			311	2.52	1.348	4
	Advanced LDT	AHTI	AHTI serve as gateways between humans and digital 311 2.30 1.282	Zouari et al. (2021)	5
	human-		tools in several fields including SCM. They are
	technology BDA		interfaces easily usable by humans that collect and 311 2.26 1.175	6
	interfaces		show data collected from other digital tools in real
	IoT		time.	311	2.14	1.237	7
	Tool Advanced SS	Abbreviation Purpose within SCM SMT 3D printing reduces the steps and materials needed to 311 2.02 1.280	Authors Schniederjans (2017) 8
	smart		manufacture small parts, reduces lead time and
	Big manufacturi data SMT	BD	Have multiple advantages in SCM: reduced transportation costs throughout the SC. 311 1.85 1.214	Ramanathan et al., 9
	analytics ng technologie ML		operational costs, improved SC agility, and increased customer satisfaction 311 1.83 1.155	(2017).	10
	s, e.g., 3D
	Artificial printing BLOCK	AI	Consists of technological platforms used to collect, 311 1.73 1.019	Russell and Norvig 11
	intelligence Location AI	LDT	analyze, store and present business data. Supports decision-making by converting raw business data to RFID provides real-time information to track the 311 1.67 1.021	(1995) Kim et al. (2008)	12
	detection technologie SDV		meaningful and valuable information and insights. movement of materials. Can be used for product forecasting, inventory and order management, 311 1.62 1.100	13
	Machine learning s, e.g., RFID VR	ML	Makes it easier to discover patterns in SC data by relying on algorithms that quickly identify the most tracking, shipping. Improves SC visibility, customer satisfaction, the timeliness of SC information. 311 1.59 1.018	Min (2010)	14
			important factors for the SCM success, while Increases profitability by reducing lead-time
			constantly learning and updating the process. variability, decreasing labor costs.
	Augmented/ Collaborativ	AR/VR COLLAB	Could be used to design a layout in a warehouse or a They support information exchanges within business	Merlino and Sproģe Romero and Vernadat
	virtual e		production line and communicate repair instructions functions and across SC actors.	(2017) (2016)
	reality technologie		through mobile or other remote-control devices.
	s (e.g., ERP,
	Mobile APS, EDI	MDW	Wrist bands, smart watches, wearable cameras or	de Assis Dornelles et
	devices/ and		smart eye-wears. Offer huge potential for increased	al., (2022)
	wearables workflow)		SC collaboration. Enable workers to access
			information and identify tasks, so improve
	Smart	SS	operational efficiencies. Are often used in warehouses Can be used in manufacturing environments, freight	Schniederjans et al.
	sensor		(order picking, tracking, inventory management). containers to detect miles, miles per gallon, fuel,	(2020)
	technology		location, speed, etc.
	Robotic	ROBOT	Is intended to replace a manual process with an	Viale and Zouari,
	process Self-driving	SDV	automated one and minimize human errors such as Fully automated transportation systems used within	(2020) Büyüközkan	and
	automation vehicles		amounts, contracts information. the industry and warehouses. It raises productivity by	Göçer (2018)
			reducing the resources needed for daily tasks.
	Internet of	IoT	Increase cost-saving, inventory accuracy, product	Gunasekaran et al.,
	Things		tracking, flexibility. IoT integrates technologies like	(2016)
	(IoT)		RFID tags, actuators.
	platforms
	Cloud	Cloud	Provides benefits such Salkin et al., (2018)
	computing						Ivanov et al., (2022)

Table IV .

 IV Digital tools adoption levels and ranking

			Mann Whitney U tests
				Construct 2: Modern decision support
	Groups	Construct 1: Operational tools	tools
		U test	Sig.	U test	Sig.
	G1 vs G2	5364.500	.002*	6000.500	.063
	G1 vs G3	2302.000	.000*	2573.500	.000*
	G2 vs G3	2381.000	.023	2231.5000	.004*

Table VII .

 VII Difference between groups regarding firm size influence on digital tools adoption -Mann Whitney tests

	Label	Challenges	N	Mean Standard	Rank
					deviation
	C5	Lack of integration	311	3.50	1.141	1
	C7	Lack of organization and skills in implementing this new	311	3.44	1.173	2
		information technology tool
	C3	Lack of knowledge -training and skills	311	3.36	1.231	3
	C1	Lack of planning	311	3.34	1.252	4
	C4	Lack of required flexible and agile SCM	311	3.33	1.219	5
	C2	Lack of collaboration between SC partners	311	3.26	1.156	6
	C8	Poor quality of data	311	3.24	1.246	7
	C6	Tendency to overestimate the potential gains	311	3.16	1.146	8
	C9	Lack of financial capacities	311	2.57	1.626	9

Table VIII .

 VIII Challenges to SC digitalization and ranking

[START_REF] Lin | Strategic response to Industry 4.0: an empirical investigation on the Chinese automotive industry[END_REF] "the nature of the company refers to the way how the company is organized". This variable was measured by the following items: State-owned joint venture, Private joint venture, Foreign companies, Domestic companies, others.

See statistics: https://ec.europa.eu/growth/smes/sme-definition_en (accessed November 5, 2021).

 Gessner, G.H., & Snodgrass, C.R. (2015)'Designing e-commerce cross-border distribution networks for small and medium-size enterprises incorporating Canadian and US trade incentive programs ', Research in Transportation Business & Management, Vol. 16, pp.84-94. Gunasekaran, A., Subramanian, N., & Tiwari, M. (2016) Technological Forecasting and Social Change, vol. 146, pp.119-132. Hughes, L., Dwivedi, Y.K., Misra, S.K., Rana, N.P., Raghavan, V., & Akella, V. (2019) 'Blockchain research, practice and policy: Applications, benefits, limitations, emerging research themes and research agenda', International Journal of Information Management, Vol. 49, pp.114-129. Hutcheson, G.D., & Sofroniou, N. (1999) 'The multivariate social Scientist: Introductory statistics using generalized linear models', Sage, London.

Challenge type Definition / Impact References

C1 Lack of planning

Internal capability is a central issue with regard to digitalization. In order to enhance this, management should focus on building roadmaps and planning strategically to invest in suitable resources.

Raj et al. (2020)

C2

Lack of collaboration

In order to support the adoption of digital tools in the SC, it is necessary that there is collaboration between the players. Some partners in SC work in their own silos, without sufficiently communicating and with significant organizational barriers, digital transformation in these situations can be a struggle. Subsequently, the ability to establish high-quality collaboration and communications on digital platforms brings organizations increased reliability, agility, and efficiency. The next generation of SCs must be stronger and more agile. As the implementation of SC digitalization is a complex process, sometimes, conflicts between workers arise due to rapid changing working environments. This has resulted in rigid organizational configurations, inaccessible data and fragmented relationships with partners. Müller, et al. (2018a);

C5

Lack of integration

The integration of SC digital tools can provide various benefits to SCM, such as a unified and whole view of inventory across the firm, the integration of customer purchase data to offer better and personalized sales and customer services. Lack of integration may hinder SC digital tools' better adoption.

Büyüközkan and Göçer (2018) C6

A tendency to overestimate the potential gains

The value and productivity gains from digital SC is unclear. The economic benefits of spending on digital tools have always been questioned, particularly for firms which lack financial resources. The adoption of new digital tools is a complex process. Moreover, it is necessary to adopt several tools simultaneously. As a result, employees and managers may resist the use of new technologies and associated practices. With a strong organizational resistance, there is a need to consider management as an essential function in the digital transformation process.

C8

Poor quality of data SC professionals are inundated with data. However, in order to make appropriate management decisions, SC managers should be assured that they have access to accurate or "good" data. The number of various tools used also makes it difficult to get aggregated and useful data, which in turn may hinder any attempt to go further with SC digital tools adoption.

Chen et al. (2014);

C9

Lack of financial

The lack and shortage of financial resources is a significant challenge. Some companies don't have enough resources to [START_REF] Kiel | The influence of the Industrial Internet of Things on business models of established manufacturing companies-A business level perspective[END_REF]


Stéphane Bessy
email: stephane.bessy@lirmm.fr

Johannes Pardey
email: johannes.pardey@uni-ulm.de

Lucas Picasarri-Arrieta
email: lucas.picasarri-arrieta@inria.fr

Dieter Rautenbach
email: dieter.rautenbach@uni-ulm.de

Factorially Many Maximum Matchings Close to the Erdős-Gallai Bound

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

We consider finite, simple, and undirected graphs. A matching in a graph G is a set of pairwise disjoint edges, and the matching number ν(G) of G is the largest size of a matching in G. For a matching M in G, let V (M ) be the set of vertices of G that are incident with an edge in M ; the set V (M ) contains the vertices of G that are saturated by M .

A classical result of Erdős and Gallai, Theorem 4.1 in [START_REF] Erdős | On maximal paths and circuits of graphs[END_REF], states that a graph G of order n, size m, and matching number ν(G) such that ν(G) = νn for some ν ∈ 0, 1 2 satisfies

m ≤ m(n, ν) :=      νn(n -νn) + νn 2 , if ν ≤ 2 5 -3 5n , and 2νn+1 2
, if 2 5 -3 5n ≤ ν ≤ 1 2 .

(1)

Furthermore, they showed that equality holds in (1) if and only if (i) the complement G of G is K n-νn ∪ K νn for ν ≤ 2 5 -3 5n , and

(ii) G is K 2νn+1 ∪ K n-2νn-1 for 2 5 -3 5n ≤ ν ≤ 1 2 .
Recall that, for positive integers n and k with k ≤ n, the falling factorial n k is n(n -1) . . . (n -k + 1).

The starting point here was the observation that the two extremal graphs in (i) and (ii) have (n -νn) νn and (2νn + 1)! (νn)!2 νn maximum matchings, respectively. Estimating quite roughly, it follows that, for positive ν, the extremal graphs for [START_REF] Brègman | Certain properties of nonnegative matrices and their permanents[END_REF] have between ⌈0.4n⌉ ⌈0.5νn⌉ and n νn maximum matchings. We show that G still has factorially many maximum matchings provided that m(n, ν) -m is sufficiently small. Since m(n, ν) = Θ(νn 2 ), it is natural to bound m(n, ν) -m in terms of ν and n 2 .

The following is our first main result; all proofs are given in the next section.

Theorem 1. For every real ν with ν ∈ 0, 1 2 , the following holds: If G is a graph of order n, size m, and matching number νn such that n ≥ 50 ν 2 and m ≥ m(n, ν) -ν 50 2 n 2 , then G has at least ⌈0.1n⌉ ⌈0.1νn⌉ maximum matchings.

For the sake of simplicity, we did not try to optimize the constants that appear in this statement, which works over the full range 0, 1 2 of ν. Our purpose here is rather to illustrate the effect and present arguments and tools that allow to capture it. In particular, the exact dependence of the minimum number of maximum matchings on the difference m(n, ν) -m remains a natural yet challenging open problem.

Our second main result gives a better bound provided that ν is sufficiently small. Theorem 2. There are two functions h ν : (0, 1) → 0, 1 2 and h δ : (0, 1) × 0, 1 2 → (0, 1) with the following property: If ϵ ∈ (0, 1), ν ∈ (0, h ν (ϵ)), and G is a graph of order n, size m, and matching num-

ber νn such that n ≥ 1/h δ (ϵ, ν) and m ≥ m(n, ν) -h δ (ϵ, ν)n 2 , then G has at least ⌈(1 -ϵ)n⌉ ⌈(1-ϵ)νn⌉ maximum matchings.
Matchings in graphs are among the most well studied topics in graph theory [START_REF] Lovász | Matching theory[END_REF], and we would like to mention only few related results. Computing the permanent of a matrix, and, hence, counting the perfect matchings of a given bipartite graph, is a well known #P-complete problem [START_REF] Valiant | The complexity of computing the permanent[END_REF]. Van der Waerden's proved conjecture on the permanent of a doubly stochastic matrix [START_REF] Egorychev | The solution of van der Waerden's problem for permanents[END_REF][START_REF] Falikman | Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix[END_REF][START_REF] Gurvits | Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all[END_REF][START_REF] Schrijver | On lower bounds for permanents[END_REF] allows to show that d-regular bipartite graphs have exponentially many perfect matchings for d ≥ 3, and Brègman's [START_REF] Brègman | Certain properties of nonnegative matrices and their permanents[END_REF] upper bound on the permanent allows to derive an exponential upper bound. Another famous related result, establishing a conjecture of Lovász and Plummer, is due to Esperet, Kardoš, King, Král, and Norine [START_REF] Esperet | Exponentially many perfect matchings in cubic graphs[END_REF] who showed that cubic bridgeless graphs have exponentially many perfect matchings.

Auxiliary results and proofs

Throughout this section, let G be a graph of order n, size m, and matching number νn.

A key tool for our approach is the Gallai-Edmonds decomposition D ∪ A ∪ C of a graph G, cf. [START_REF] Lovász | Matching theory[END_REF],

where

D = {u ∈ V (G) : ν(G -u) = ν(G)}, A = u∈D N G (u) \ D, and C = V (G) \ (D ∪ A),
that is, the set D contains the vertices that are not saturated by some maximum matching in G, the set A is the set of neighbors of the vertices in D outside of D, and C contains the remaining vertices.

Let the components of G

[D] be G 1 , . . . , G k . Each G i is factor-critical, that is, for every vertex u of G i , the graph G i -u has a perfect matching. Let G i have order n i for i ∈ [k], d = |D|, a = |A|, and c = |C|.
Every maximum matching of G consists of, cf. [START_REF] Lovász | Matching theory[END_REF],

• a matching of size

(n i -1)/2 in G i for every i ∈ [k],
• a matching of size a in the bipartite subgraph of G with the partite sets A and D formed by the edges between these two sets, and

• a perfect matching in G[C].
In particular, such matchings are guaranteed to exist. Note that

n -2νn = k -a (2)
is the number of vertices that are not saturated by maximum matchings in G.

If G * arises from G by adding all missing edges

• within V (G i ) for each i ∈ [k],
• between D and A, and

• within A ∪ C,
then G * has the same Gallai-Edmonds decomposition, and, hence, also the same matching number as G, and

m(G * ) = k i=1 n i 2 + da + n -d 2 . ( 3
)
Using d = n 1 + . . . + n k , n 1 , . . . , n k ≥ 1, and the convexity of x → x 2 , we obtain m(G * ) ≤ m * := d -k + 1 2 + da + n -d 2 . (4)
Since (4) holds with equality if each but at most one component of G * is an isolated vertex, the integer m * is the size of a graph of order n and matching number νn, and (1) implies m(n, ν) ≥ m * .

We introduce the useful variables x and y:

x = ν - a n and y = d -k n . (5)
Since 0 ≤ a ≤ νn, we have x ∈ [0, ν].

Note that

k (2) = n -2νn + a = n -2νn + νn -xn = (1 -ν -x)n, (6)
and, hence,

y = d -k n ≤ n -a -k n = 1 -ν + x -1 + ν + x = 2x, that is, y ∈ [0, 2x].
We provide some intuition for x and y. The variable x measures which proportion of every Our first lemma expresses the quadratic part of m(n, ν) -m * in terms of x and y.

Lemma 3. m(n, ν) -m * ≥ g(x, y)n 2 -n for g(x, y) :=    x 1 -ν -3 2 x + y(2x -y) , if ν ≤ 2 5 -3 5n and (ν -x) 5 2 ν + 3 2 x -1 + y(2x -y) , if 2 5 -3 5n < ν ≤ 1 2 . Proof. If ν ≤ 2 5 -3 5n , then m(n, ν) -m * (1),(4) = νn(n -νn) + νn 2 - d -k + 1 2 + da + n -d 2 = νn(n -νn) + (νn) 2 2 - (d -k) 2 2 + da + (n -d) 2 2 - νn 2 + d -k 2 - n -d 2 ≤n ≥ νn(n -νn) + (νn) 2 2 - (d -k) 2 2 + da + (n -d) 2 2 -n (5) = x 1 -ν - 3 2 x + y(2x -y) n 2 -n,
where the last equality requires a tedious yet straightforward calculation.

Similarly, if 2 5 -3 5n < ν ≤ 1 2 , then m(n, ν) -m * (1),(4) = 2νn + 1 2 - d -k + 1 2 + da + n -d 2 = 2(νn) 2 - (d -k) 2 2 + da + (n -d) 2 2 --νn + d -k 2 - n -d 2 ≤n ≥ 2(νn) 2 - (d -k) 2 2 + da + (n -d) 2 2 -n (5) = (ν -x) 5 2 ν + 3 2 x -1 + y(2x -y) n 2 -n,
where the last equality again requires a tedious yet straightforward calculation.

The next lemma allows to identify the values of x and y for which m(n, ν) -m * is small. Before we formulate it, we motivate its statement using plots of g(x, y) for different values of ν, cf. Figure 1.

For ν = 0.4, the function g(x, y) is small in three cases:

(1) Either x, and, hence, also y are small, (2) or x is close to ν, and y is small, (3) or x is close to ν, and y is close to 2ν.

For smaller values of ν, for instance, ν = 0.

3 n . If x ≥ δ ∨ ν ≥ 2 5 + 2 5 δ ∧ x ≤ ν -δ ∨ ν ≤ 2 5 - 2 5 δ ∨ δ ≤ y ≤ 2ν -3δ , (7)
then g(x, y) ≥ δ 2 .

Proof. First, we assume that ν ≤ 2 5 -3 5n , which implies x ≥ δ.

If x ≤ ν -δ, then g(x, y) = x 1 -ν - 3 2 x + y(2x -y) ≥0 ≥ x ≥δ 1 -ν - 3 2 x ≥ 3 2 δ ≥ δ 2 , if x > ν -δ and ν ≤ 2 5 -2 5 δ, then g(x, y) = x 1 -ν - 3 2 x + y(2x -y) ≥0 ≥ x ≥δ 1 -ν - 3 2 x ≥δ ≥ δ 2 ,
and, if x > ν -δ and ν > 2 5 -2 5 δ, then

g(x, y) = x 1 -ν - 3 2 x ≥0 +y(2x -y) ≥ y ≥δ (2x -y) ≥δ ≥ δ 2 .
Next, we assume that 2 5 -3 5n < ν ≤ 1 2 , which implies ν > 2 5 -2 5 δ. Note that

5 2 ν + 3 2 x -1 ≥ min 5 2 2 5 - 3 5n + 3 2 δ -1, 5 2 2 5 + 2 5 δ -1 ≥ δ. If x ≤ ν -δ, then g(x, y) = (ν -x) 5 2 ν + 3 2 x -1 + y(2x -y) ≥0 ≥ (ν -x) ≥δ 5 2 ν + 3 2 x -1 ≥δ ≥ δ 2 ,
and, if x > ν -δ, then

g(x, y) = (ν -x) 5 2 ν + 3 2 x -1 ≥0 +y(2x -y) ≥ y ≥δ (2x -y) ≥δ ≥ δ 2 .
The next two lemmas establish the existence of many maximum matchings in graphs that are close to complete bipartite graphs or complete graphs, respectively.

Lemma 5. For positive integers k and a with k > 1.1a, let the bipartite graph H arise from K a,k with partite sets A and K of orders a and k, respectively, by removing up to 0.08a 2 edges in such a way that H has a matching saturating all vertices in A.

The graph H has at least ⌈k -1.1a⌉ min{⌈0.2a⌉,⌈k-1.1a⌉} matchings saturating all vertices in A.

Proof. Let M be a matching in H saturating all vertices in A.

If A contains a subset A ′ of at least 0.2a vertices u with |N H (u) \ V (M )| ≥ k -1.
1a, then there are at least ⌈k -1.1a⌉ min{⌈0.2a⌉,⌈k- 1.1a⌉} matchings that connect the vertices in A ′ to vertices in K \ V (M ). Since each of these matchings can be extended to a matching saturating all vertices in A by using edges from M , the desired statement follows. Hence, for a contradiction, we suppose that a set A ′ as above does not exist. Since H has at most a 2 edges within V (M ), this implies that m(H) < a 2 + 0.2a(k -a) + 0.8a(k -a -0.1a) = ak -0.08a 2 , which is a contradiction. Lemma 6. For a positive integer p with p ≥ 10 3 , let the graph K arise from K 2p by removing up to 0.01 p 2 many edges in such a way that K has a perfect matching. The graph K has at least ⌈0.447p⌉! perfect matchings.

Proof. Let M = {x 1 y 1 , . . . , x p y p } be a perfect matching in K. Let the graph H with vertex set {z 1 , . . . , z p } be such that, for distinct indices i and j from [p], the two vertices z i and z j are adjacent in H if and only if the two edges x i y j and x j y i belong to K. Since every non-edge in K leads to at most one non-edge in H, we have

m(H) ≥ p 2 -0.01 p 2 ≥ 0.99 p 2 p≥10 3 ≥ 0.989 p 2 2 ,
which implies that H has average degree at least 0.989p. This implies that H contains at least 0.895p vertices of degree at least 0.895p. Hence, the graph H contains a set X of at least 0.447p vertices such that each vertex in X has at least 0.447p neighbors outside of X. This immediately implies that H contains at least ⌈0.447p⌉! distinct matchings.

For a matching N in H, let M (N ) arise from the perfect matching M in K by replacing, for every edge z i z j in N , the two edges x i y i and x j y j in M with the two edges x i y j and x j y i . Clearly, the set M (N ) is a perfect matching in K, and M (N ) is distinct from M (N ′ ) for distinct matchings N and N ′ in H. Therefore, the graph K has at least ⌈0.447p⌉! perfect matchings.

Proof of Theorem 1. Let ν ∈ 0, 1 2 . Let

ϵ = ν 50 2 ≤ 10 -4 . ( 8
)
Let G be a graph of order n, size m, and matching number νn such that ϵn ≥ 1 and m ≥ m(n, ν)-ϵn 2 .

We have

ϵn 2 ≥ m(n, ν) -m = (m(n, ν) -m * ) ≥0 + (m * -m(G * )) ≥0 + (m(G * ) -m) ≥0 , (9)
where we use the notation introduced earlier in this section.

Let δ = 2 √ ϵ.
Using (8), ν ≤ 1 2 , and n ≥ 1 ϵ , we obtain

0 < δ ≤ ν 25 , and n ≥ max 3 δ , 10 4 . (10)
If ( 7) holds, then Lemma 3 and Lemma 4 imply

ϵn 2 (9)
≥ m(n, ν) -m * ≥ g(x, y)n 2 -n ≥ δ 2 n 2 ≥4ϵn 2 -n ≤ϵn 2 ≥ 3ϵn 2 ,
which is a contradiction. Hence (7) fails, which leads to the following three cases:

• x < δ and ν < 2 5 + 2 5 δ.

• x > ν -δ, ν > 2 5 -2 5 δ, and y < δ.

• x > ν -δ, ν > 2 5 -2 5 δ, and y > 2ν -3δ.

Each of these three cases corresponds to a different degeneration of the Gallai-Edmonds decomposition of G. We now consider these cases separately.

Case 1 x < δ and ν < 2 5 + 2 5 δ.

Note that ν < 2 5 + 2 5 δ (10)
≤ 2 5 + 2 125 ν, which implies ν < 0.407. (11)
By ( 5) and ( 6), we have

νn ≥ a = (ν -x)n > (ν -δ)n (10)
≥ 0.96νn, and

(1 -ν)n ≥ k (6) = (1 -ν -x)n > (1 -ν -δ)n (10), (11)
> 0.576n.

Let H 0 be the bipartite subgraph of G with the partite sets A and D formed by the edges between these two sets. Let M be a matching of size a in H 0 that is a subset of some maximum matching in G. In particular, for every i ∈ [k], there is at most one edge in M between A and V (G i ). Let H arise from H 0 by removing, for every component G i of G[D], all but exactly one vertex in such a way that

V (M ) ⊆ V (H).
The properties of the Gallai-Edmonds decomposition imply that every matching in H that saturates all vertices in A can be extended to a maximum matching in G. By (9), we have > 0.1n ≥ 0.1νn and 0.2a

m(G * ) -m ≤ ϵn 2 ,
> 0.1νn, the desired statement follows in this case.

Case 2 x > ν -δ, ν > 2 5 -2 5 δ, and y < δ.

We have ν > 2 5 -2 5 δ

(10)

≥ 2 5 -2 125 ν, which implies ν > 0.393. ( 13
)
Furthermore,

|C| = n -d -a = n -(d -k) -k + (νn -a) -νn (5),(6)
= (1 -y -1 + ν + x + x -ν) n = (2x -y)n > (2ν -3δ)n (10)
≤ ϵn 2 many edges. Since 0.01 p 2 ≥ 0.01•0.368

2 2 n 2 (8)
≥ ϵn 2 , and every perfect matching in G[C] can be extended to a maximum matching in G, Lemma 6 implies that G has at least ⌈0.447p⌉! ≥ ⌈0.1n⌉! maximum matchings, and the desired statement follows in this case.

Case 3 x > ν -δ, ν > 2 5 -2 5 δ, and y > 2ν -3δ.

Exactly as in Case 2, we obtain (13). Furthermore,

|D| ≥ |D| -k = d -k (5)
= yn > (2ν -3δ)n

(10) ≥ 2 - 3 25 νn (13)
≥ 0.738n.

Suppose, for a contradiction, that max{n 1 , . . . , n k } ≤ 0.54n. Using the convexity of x → x 2 , we obtain

ϵn 2 (9) ≥ m * -m(G * ) (3),(4) = d -k + 1 2 - k i=1 n i 2 > (d -k) 2 2 - 1 2 k i=1 n 2 i ≥ 1 2 (0.738n) 2 - 1 2 n 0.54n (0.54n) 2 ≥ 0.002n 2 , contradicting (8)
. Hence, we may assume that n 1 ≥ 0.54n

≥ 10 3 . For p = n 1 -1 2 , we obtain p ≥ 0.269n ≤ ϵn 2 many edges. Furthermore, by the properties of the Gallai-Edmonds decomposition, the graph K has a perfect matching, and every perfect matching in K can be extended to a maximum matching in G. Since 0.01 p 2 ≥ 0.01•0.268 2 2 n 2

≥ ϵn 2 , Lemma 6 implies that G has at least ⌈0.447p⌉! ≥ ⌈0.1n⌉! maximum matchings, and the desired statement follows in this case.

This completes the proof.

For the proof of Theorem 2, we need the following variant of Lemma 5.

Lemma 7. Let ϵ, γ ∈ (0, 1). For positive integers k and a with ϵk 2 ≥ a, let the bipartite graph H arise from K a,k with partite sets A and K of orders a and k, respectively, by removing up to γak edges in such a way that H has a matching saturating all vertices in A.

The graph H has at least 1

- √ γ -ϵ 2 k ⌈(1- √ γ-ϵ
2 )a⌉ matchings saturating all vertices in A.

Proof. Let M be a matching in H saturating all vertices in A.

If A contains a subset A ′ of at least 1 - √ γ a vertices of degree at least 1 - √ γ k, then each vertex u in A ′ satisfies |N H (u) \ V (M )| ≥ 1 - √ γ k -a ≥ 1 - √ γ -ϵ 2 k, and there are at least 1 - √ γ -ϵ 2 k ⌈(1- √ γ-ϵ
2 )a⌉ matchings that connect the vertices in A ′ to vertices in K \ V (M ). Since each of these matchings can be extended to a matching saturating all vertices in A by using edges from M , the desired statement follows.

Hence, for a contradiction, we suppose that a set A ′ as above does not exist. This implies that m(H) < 1 -√ γ ak + √ γa 1 -√ γ k = ak -γak, which is a contradiction.

Proof of Theorem 2. Let ϵ ∈ (0, 1). Let ν ∈ 0, 1 2 and δ ∈ (0, 1) be such that

Further restricting ν (in terms of ϵ) and δ (in terms of ϵ and ν), we assume that

1 -1 + ϵ 2 ν ≥ 1 - ϵ 2 , and 1 - δ 1 -ϵ 2 ν 1 -1 + ϵ 2 ν - ϵ 2 1 - ϵ 2 ≥ 1 -ϵ. ( 15
)
Let G be a graph of order n, size m, and matching number νn such that δn ≥ 1 and m ≥ m(n, ν)-δn 2 .

Using the notation introduced before in this section, Lemma 3 implies 2δ ≥ 1 n 2 (m(n, ν) -m * + n) ≥ g(x, y) ≥ x 1 -ν -

3 2 x ≥ x 1 - 5 2 ν ≥ x 2 .
Hence, x ≤ 4δ, and νn ≥ a = (ν -x)n ≥ (ν -4δ)n

1 - √ γ - ϵ 2 k (16) ≥ 1 - √ γ - ϵ 2 1 -1 + ϵ 2 ν n (15) ≥ 1 - √ γ - ϵ 2 1 - ϵ 2 n (15)
≥ (1 -ϵ)n, and

1 - √ γ - ϵ 2 a (16) ≥ 1 - √ γ - ϵ 2 1 - ϵ 2 νn (15)
≥ (1 -ϵ)νn, which completes the proof.

  maximum matching consists of edges between A and D. If x = 0, then a = νn, which means that every maximum matching lies completely between A and D. If x = ν, then a = 0, which means that every maximum matching lies completely within D ∪ C. The variable y relates to the orders of the components of G[D]. Each of these k components contains at least one vertex, that is, d -k is the number of additional vertices of G distributed among the components of G[D] on top of this one vertex. Therefore, y measures the proportion of the vertices of G that fill up the k components of G[D]. If y = 0, then all components of G[D] have just one vertex. If y = 2x, then n = a + d, which means that C is empty.

Figure 1 :

 1 Figure 1: The function g(x, y) for ν = 0.3 (left), ν = 0.4 (middle), and ν = 0.5 (right), assuming that n is large enough.

  |C| = 2p, then p ≥ 0.369n (10) ≥ 10 3 , and, hence, p -1 ≥ 0.368n. Note that G[C] arises from K 2p by removing at most m(G * ) -m

≥

  10 3 and p -1 (10) ≥ 0.268n. Note that removing any one vertex from G 1 , we obtain a graph K that arises from K 2p by removing at most m(G * ) -m[START_REF] Valiant | The complexity of computing the permanent[END_REF]

ϵ 2 ,

 2 and δ ≤ ϵν 8.

= ( 1 -

 1 ν -x)n ≥ (1 -ν -4δ)n upper bound on ν in (14), this implies ϵ 2 k ≥ a. Let the graph H be defined as in Case 1 of the proof of Theorem 1.For γ = δ (1-ϵ 2 )ν(1-(1+ ϵ 2 )ν), we obtain γak

≥⌈( 1 -√ γ-ϵ 2 )

 12 δn 2 , and Lemma 7 implies that H, and, hence, also G, has at least 1 -√ γ -ϵ 2 k a⌉ maximum matchings. Now,

  and, hence, the graph H arises from K a,k with partite sets A and K of orders a and k, respectively, by removing up to ϵn 2 edges in such a way that ν(G) = a. Since k

			(11),(12)
			> 1.1a and
	0.08a 2	(12) ≥ 0.08 • 0.96 2 ν 2 n 2	(8) ≥ ϵn 2 , Lemma 5 implies that H has at least ⌈k -1.1a⌉ min{⌈0.2a⌉,⌈k-1.1a⌉}
			(11),(12)
	matchings saturating all vertices in A. Since k -1.1a

Research supported by research grant DIGRAPHS ANR-19-CE48-0013.