P Gateau

C Petitjean

P J Panteix

C Rapin

M Vilasi

Solubility of tin dioxide in soda-lime silicate melts

Keywords: glass melts, tin dioxide, solubility

In the present work, the solubility of tin dioxide is assessed as a function of time, temperature and basicity in simple ternary glasses: NC3S, NC4S, NC5S and NC6S (N: Na2O, C: CaO, S: SiO2). An increase of silica contents in the glass composition leads indeed to a decrease of the glass basicity. First, a kinetic study of the dissolution has been performed. Consequently, the solubility limits of tin dioxide have been determined after 2 hours of heat treatment: this duration is long enough to reach the dissolution equilibrium, and short enough to limit the sodium oxide losses in the melt at high temperatures. Nevertheless the specific case of the most acid glass has been underlined, as its higher viscosity implies longer heating times. At equilibrium state, SnO2 solubility depends on the temperature (Arrhenius law) and on the glass basicity. In the 1200°C-1400°C temperature range, in these soda-lime glasses, the solubility of tin dioxide is between 1.3 and 2.1 at.%Sn and the temperature dependence of solubility exhibits a single mechanism of dissolution. Furthermore, the basicity dependence of the solubilization process is also discussed, and the presence of another oxidation state of tin (Sn II ) is thus proposed.

Introduction

Many applications in glass industry require high performance materials, with good mechanical properties and high resistance to corrosion at high temperature. Several ceramic materials, as tin dioxide, are used in the molten glass media [1]. Cassiterite, SnO2, is already used for specific applications such as heating electrodes, and high temperature devices [2][START_REF] Guigonis | Use Of Sintered Refractory Materials Based On Tin Oxide For Producing Glass Furnace Throats[END_REF][4][5].

Soda-lime glasses are 90% of the industrial glass production [START_REF] Barton | Le Verre[END_REF]. The lifetime of ceramic materials strongly depends on the corrosion rate of this material in high temperature conditions. In silicate melts, this corrosion rate is directly linked to the stability of the oxide, i.e. to its solubility in molten glass. As a consequence, a more accurate knowledge of the oxide dissolution in a simple silicate melt will provide useful information regarding the ability of this ceramic to remain (chemically) inert in glass melts.

The aim of this work is to study the influence of glass composition on the solubility of SnO2 in simple ternary glasses Na2O-CaO-xSiO2 (with x = 3, 4, 5 and 6; noted as NCxS in this paper). These glasses exhibit low viscosity values at temperatures higher than 1200°C [START_REF] Neuville | [END_REF]. As a consequence, in order to get thermodynamic data about the dissolution equilibrium, the temperature range chosen in this work rises from 1200°C to 1400°C. In a first time, kinetic aspect has been considered to estimate the equilibrium time of the dissolution reaction. In a second time, the limit of solubility (once the equilibrium is reached) has been determined at different temperatures (from 1200°C to 1400°C). The solubility limits have also been discussed as a function of glasses basicity.

Experimental procedure

At first, four ternary glasses of general composition Na2O-CaO-xSiO2 (x=3, 4, 5, and 6) were prepared with the standard procedure settled by .

The raw chemicals reagents NaCO3 (99.5 %), CaCO3 (99 %), and amorphous SiO2 (99.8 %), were mixed and grinded in a planetary ball mill during 15 minutes. The homogenized mixture was charged in a Pt -10%Rh crucible and heated. The temperature has been raised slowly (10 hours) up to 1200°C in order to enable progressive decomposition of the carbonates. It has been maintained at 1500°C during 2 hours to ensure a good homogenization of the melt, and then at 1300°C for 10 hours. The glass was finally quenched in air at 20°C to avoid glass crystallization. Glasses compositions were determined by Electron Probe MicroAnalysis (EPMA) and are reported in Table 1, as well as the theoretical compositions. The ratios x' (2at.%Ca/at.%Na) and x (average value of at.%Si/at.%Ca and 2at.%Si/at.%Na) are estimated from these measurements. The calculated optical basicities th of the glasses are also reported.

In all the prepared glasses, the atomic ratio 2Ca/Na (x') is very close to 1, allowing the NCxS notation. Furthermore, the x ratio is close to the expected theoretical glass stoichiometry. As a consequence, these results validate the process used here for the synthesis of ternary glasses.

In a second time, the solubility of SnO2 has been studied following the protocol previously described by Khedim et al. [8][9][10]. The process is summarized on Fig. 1. Twenty grams of each glass were crushed with cassiterite powder (SnO2 99.8 %, Oximet); an amount of 20 wt.% has been chosen in order to be higher than the solubility limit of SnO2. This value has been chosen knowing that Manfredo et al. have measured limits of solubility of about 16

wt.% at 1500°C in a common soda-lime glass [14]. Mixtures were heated 2 minutes at 1200°C, and then quenched. It was done twice to ensure good homogeneity of the glass/oxide mix. Pieces of 100 mg of SnO2-enriched glasses (noted here as SnO2-NCxS) were placed in graphite crucibles at 1200°C for 2 min. Due to graphite's low wettability regarding glass, the samples formed balls with an average diameter of 4 mm. Balls, sustained by a Pt-wire in alumina crucible, were then annealed at temperatures comprised between 1200°C and 1400°C, for times up to 9 hours in air.

Quenched samples were embedded in epoxy resins for metallographic surface preparation ending with polishing using 1 µm diamond suspension. SnO2 over-saturation was controlled by Scan Electron Microscope (SEM) using a BSE detector with an acceleration voltage of 15 kV. The polished sections were then analyzed with a CAMECA SX 100 electron microprobe in order to determine the total amount of tin dissolved in the glassy matrix. Selected elements for analysis were Na, Si, Ca and Sn. The concentration of oxygen was calculated by stoichiometry assuming that the valence states of the cations are +I, +IV, +II and +IV respectively. The standards used were albite NaAlSi3O8 for Na and Si, andradite Ca3Fe2(SiO4)3 for Ca, and cassiterite SnO2. Specific conditions were optimized to limit sodium migration under the electron beam [11]: the acceleration voltage was set at 15 kV, beam current at 10 nA, with time count of 10 s and electron beam defocalized to a 20 µm diameter. Measurements have been performed in the vicinity of SnO2 crystals to estimate the tin content dissolved in the glass matrix. Such conditions ensure the measurement of the highest tin content in the glass to determine the solubility limit. The results are estimated from 6 to 8 analyses, and error-bars correspond to standard deviations.

Results

Tin dioxide contents measured in all samples after heat treatments at 1200°C, 1250°C, 1300°C and 1400°C for times of 1h, 2h, 4h and 9h are reported in Table 2. For each test, the samples were first observed by SEM. Fig. 2 presents micrographs of a glass enriched with SnO2 and heated at several temperatures for different times. The results presented here only deal with NC5S, but similar observations have been performed in all the glasses studied here.

The micrographs reveal the presence of remaining tin dioxide grains (white contrast: confirmed by EDX analysis shown on Fig. 2d) in the glass matrix (dark contrast). For short times and low temperatures (e.g. 1h at 1200°C; Fig. 2a), tin dioxide grains are similar to the ones observed in the initial glass balls: SnO2 powder is finely dispersed in the glass and the dissolution process has just been initiated. Few bigger grains indicate a slight coalescence.

This coalescence seems to be thermally activated, as more aggregates are observed at higher temperature (e.g. at 1300°C; Fig. 2b). The content of dissolved tin and the SEM observations show that dissolution and coalescence may be in competition. Fig. 2c displays the most extreme conditions: highest temperature and longest time (1400°C for 9 hours). SnO2 grains of diameter between 1 and 20 µm exhibit a geometrical and angular shape. This particular morphology suggests that these grains are no more resulting from coalescence but from recrystallization.

The study of the solubility limit of SnO2, in several glasses at different temperatures, requires foremost determining the time necessary to reach the dissolution reaction's equilibrium. This reaction is an acid-base one and is detailed in Eq. ( 1).

- - - - +  2 ) ( ) 4 2 ( ) ( ) ( 2 ) 2 ( melt z melt z IV s IV O z O Sn O Sn (1)
The kinetics of dissolution of SnO2 at 1200°C and 1400°C for the most basic glass (NC3S)

and the most acid glass (NC6S) are reported on Fig. 3a and 3b respectively. At 1200°C, for both glasses, the content of dissolved tin reaches a constant value for times greater than 2 hours. Using another process, Manfredo et al. [14] determined the limit of solubility of tin dioxide after 3 days of heat treatment. The protocol used in the present work requires much shorter times. It can be noticed that the glass containing the lowest amount of silica (NC3S), the most basic one, is in the equilibrium state after only 1 hour. In the same way, the glass with the highest content of silica (NC6S), the most acid one, seems to require a longer heating time to reach equilibrium.

At higher temperature (1400°C, Fig. 3b), the equilibrium state is reached for shorter times (less than 1h), independently from the glass composition. Nevertheless, in the case of NC3S, the limit of solubility tends to decrease for the longest times of heat treatment at 1400°C, whereas it tends to remain constant for the NC6S sample. Furthermore, tin dioxide presents a higher solubility in the most acid glass (NC6S) than in NC3S. Lastly, these results exhibit a great influence of the temperature on the solubility limit, which is almost twice higher at 1400°C than at 1200°C, independently from the glass composition.

Discussion

Thermo-kinetic aspect

As described previously, regarding Fig. 3a and 3b, the limit of solubility of SnO2 in NC3S seems to decrease with time (this trend is confirmed by the tight error bars), especially for the highest temperatures (Fig. 3b). As this phenomenon is mainly observed in the glass with the highest alkali content, a modification of the glass composition of the glass by volatilization of sodium is proposed. The volatility of sodium is a quite common phenomenon in melts [START_REF] Konta | Volatility of oxides from silicate melt and the origin of moldavites[END_REF].

As the silicon content is known to be stable with temperature, evolutions of the ratio Na2O/SiO2 with time at 1200°C and 1400°C are reported for the glasses with the highest (NC3S) and the lowest (NC6S) sodium content on Fig. 4a and 4b respectively. Sodium losses in NC3S are clearly observed on Fig. 4a. At 1400°C the Na2O/SiO2 ratio decreases from 0.33 to 0.26 in 9 hours. Lower temperatures lead to lower sodium losses. During the same time the Na2O/SiO2 ratio in NC6S remains constant regardless of the temperature (Fig. 4b) and is in good agreement with the theoretical ratio and the one estimated from the EPMA results listed in Table 1. Furthermore, the lower solubility, linked to the sodium losses, can be correlated to the re-crystallization of tin dioxide grains observed in Fig. 2c for long times of heat treatment at the highest temperature (1400°C).

The aim of these kinetic considerations was to choose the duration of heat treatment to achieve equilibrium state. Considering these aspects, if the sodium volatilization is taken into account for the longest times and for the highest temperatures in sodium-rich glasses, it is necessary to choose a time as short as possible. Therefore 2 hours of heat treatment seems to be a good compromise.

From this, it is possible to consider the evolution of the solubility values for Na2O-CaO-xSiO2

glasses after 2 hours at 1200°C, 1250°C, 1300°C and 1400°C. The logarithm of the amount of dissolved tin dioxide is plotted as a function of the inverse of absolute temperature in Fig. 5 for NC3S, NC4S and NC5S. Both parameters are correlated, according to Arrhenius law (Eq.

(

)):

C T R E x a Sn ln 1 ) ln( +       - = ( 2
)
where Ea is the activation energy of SnO2 dissolution, R the universal gas constant, T the absolute temperature in K and C a constant.

In NC3S, NC4S and NC5S, figure 5 shows a linear evolution of ln(at.%SnO2) vs 1/T, with a correlation coefficient higher than 0.99. As a consequence, it is assumed that there is a single dissolution mechanism. The activation energies are reported in table 2. Moreover, the activation energy increases with the glass basicity, showing a higher temperature dependence of the solubility in the most basic glasses. Manfredo et al. [14] studied the solubility of tin dioxide in a commercial soda-lime glass which optical basicity (0.625) is similar to the optical basicity of NC6S studied here (0.629). From their measurements performed in the 1300°C-1600°C range, an activation energy Ea = 32 kJ.mol -1 can be calculated. This is in very good agreement with the ones determined in this study and allows the extrapolation of the single mechanism of solubilization at temperatures up to 1600°C.

The temperature dependence of tin dioxide solubility is also presented in Arrhenius format in the case of NC6S (Fig. 6). Regarding NC6S, the quality of the linear fit (R²  0.95) is much lower than for the other glasses (Fig 6a). The low linearity of the Arrhenius plot of NC6S is confirmed when including further data in the plot: the solubility measurements have also been performed for 2 hours of heat treatment at 1325°C, 1350°C and 1375°C. The temperature dependence of tin dioxide solubility in NC6S has been plotted for measurements performed after 9 hours of heat treatment (Fig. 6b). The quality of the linear fit is significantly improved, with R² higher than 0.99, which is comparable to the other glasses after 2 hours of heat treatment. NC6S is the most acid glass among those studied here, with the highest silica content. This implies that its network is not as opened as for the other glasses, and contains less non-bridging oxygens, leading to a viscosity significantly higher. As a consequence, the time required to reach equilibrium state of dissolution might be longer in NC6S than in the more basic glasses. The same kind of behavior has been observed in previous studies dealing with the dissolution of chromium oxide Cr2O3 in binary glasses [11].

The amount of dissolved tin measured after 2 hours and 9 hours of heat treatment are the same if the standard deviations are taken into account (Table 2). Further observations of Fig. 3a and 3b show for both temperatures (1200°C and 1400°C), in the case of NC6S, that the errors bars are much larger for short times and become thinner after 9 hours of heat treatment. On the other hand, for NC3S, the error bars are always quite thin, for all temperatures and heat treatments durations. As measurements are always made in the vicinity of the remaining SnO2 crystals, this means that longer times are required to reach a homogeneous repartition of tin in this area in the case of NC6S. As a consequence, the highest quality of the results (thinner error bars, linear fit of higher quality) suggests to rather considering the measurement performed after 9h in the case of the most acid glass. The specific case of NC6S allows this, as the composition of the glass remain constant for long times of heat treatment, without any measured sodium loss, even at 1400°C (Fig. 4b). Lastly, it can be noticed that the activation energies determined after 2 hours and 9 hours of heat treatment for NC6S are quite the same (Table 2), describing a single dissolution mechanism in both cases.

Influence of basicity

Several authors have previously shown that glasses with high basicity tend to stabilize the higher degree of oxidation of the cations dissolved in the melts [15,16]. If, in the case of the most basic glass studied here (e.g. NC3S), we consider that tin is dissolved as a Sn IV oxocomplex (Eq. ( 1)), the decrease of SnO2 solubility (Fig. 3b) with time can be explained by the simultaneous sodium volatilization (Fig. 4b): decreasing basicity of NC3S does not favor the stabilization of the oxidized state of tin i.e. Sn IV . The same trend has clearly been observed by Benne et al. [16] in the case of soda-lime glasses with various sodium contents.

It has also been previously reported that in silicate melts, Sn IV presents an octahedral coordination and is thus surrounded by 6 oxygen atoms [16,17]. Considering Eq. ( 1), z should be equal to 6, imposing the presence of O 2-ions on the left side of the equation (Eq. ( 3)).

--

 + 8 ) ( 6 2 ) ( ) ( 2 4 melt IV melt s IV O Sn O O Sn ( 3
)
As a consequence, the dissolution of tin dioxide into Sn IV oxocomplexes with octahedral environment should increase with the melt basicity.

It can be of great interest to plot the evolution of tin dioxide solubility as a function of the glasses basicity. In this work, amongst several models, the optical basicity  of oxide glasses is determined according to the model proposed by Duffy and Ingram Eq. ( 4) [18]:

i i oxide melt x    =  ( 4
)
where xi is the molar fraction of oxide i and i the optical basicity measured by Duffy and Ingram (SiO2=0.48, Na2O=1.15, CaO=1). The optical basicities of all NCxS glasses were calculated from their compositions measured by EPMA after 2 hours of heat treatments (9 hours for NC6S) and are reported in Table 1. The plot of the amount of dissolved tin versus optical basicity is reported in Fig. 7 for several temperatures. For all the temperatures studied here, solubility is clearly decreasing when the basicity is increasing.

As the composition of the raw glasses becomes more acid, the speciation of Sn must also be taken into account. Indeed, depending on the redox conditions of the system, Sn IV complexes can be reduced into Sn II complexes, according to the redox reaction:

II IV Sn e Sn  + - 2 Reduction of Sn IV (5)
- - +  e O O gaz 2 2 1 ) ( 2 2 Oxidation of oxide ion (6) ________________________________________________ ) ( 2 2 2 1 gaz II IV O Sn O Sn +  + - Global reaction (7) ) ( 2 ) 2 2 ( ) ( 2 ) ( ) 4 2 ( ) ( 2 1 ) 1 ( gaz
y melt y II melt z melt z IV O O Sn O z y O Sn +  - + + - - - - - (8)
A schematic summary of all reactions occurring when SnO2 is in molten glass is given in Fig. 8. First, the dissolution of tin dioxide into Sn IV complexes has been described above in Eq.

(1). It is an acid-base reaction which is independent of oxygen fugacity fO2 (i.e. redox of the glass). Then, the reduction of Sn IV into Sn II given by Eq. [START_REF] Neuville | [END_REF] or Eq. ( 8) depends on both O 2- activity and oxygen fugacity.

In the same way, the reduction of Sn IV into Sn II can be considered as the basicity decreases from NC3S to NC6S glass. EPMA measurements give access to total tin dissolved in the glass, independently from the oxidation state (Eq. ( 9)): Snmeasured = Sn II +Sn IV (9) As a consequence, the increase of tin solubility with glass acidity for a given temperature when equilibrium state is reached (Fig. 7), can be explained by the stabilization of an increasing amount of Sn II complexes.

It could be of great interest to quantify the tin species in the silicate melts. As shown in Eq. ( 7) and ( 8), the control of the redox in a glass is possible through oxygen fugacity. An experimental protocol, as the one developed by Mathieu et al. [19], should be used for this, as it has already been employed to determine the solubility of oxides with multivalent cations such as chromium oxide [8][9][10].

Conclusions

Several important features of tin chemistry in silicate melts were drawn up in this study. Among these, some of them must be underlined:

-The development of an experimental procedure led to the determination of tin dioxide solubility limits in soda-lime silicate glasses as a function of temperature and glass composition.

-The equilibrium state has been determined (kinetic considerations). Furthermore, the volatilization of alkaline oxides has been taken into account and thus limited through optimized short heat treatments (2 hours). Nevertheless, in the case of the most acid glasses, longer times (up to 9 hours) must be considered to reach the equilibrium state, due to a higher viscosity of these glasses -For a given glass and heat treatments long enough to reach equilibrium state, tin solubility displays an Arrhenius temperature dependence.

-This study reveals a correlation between the basicity of the glasses and the solubility of tin dioxide. Globally, the solubility of tin dioxide is enhanced in more acidic glasses (silica-rich glasses). Indeed, the presence of Sn II is suspected as the acidity of the glasses increases and the total solubility of SnO2 might be assimilated to the sum of dissolved Sn II and Sn IV . As a consequence, future works should deal with tin dioxide solubility in controlled atmosphere with various oxygen fugacities, in order to stabilize the different oxidation states of tin.

Figures captions.

Fig. 1 .

 1 Fig. 1. Process used for the preparation of the samples for the study of the solubility of tin oxide in silicate melts.

Fig. 2a .

 2a Fig. 2a. Micrograph of NC5S after 1h at 1200°C (BSE, 15kV).

Fig. 2b .

 2b Fig. 2b. Micrograph of NC5S after 2h at 1300°C (BSE, 15kV).

Fig. 2c .

 2c Fig. 2c. Micrograph of NC5S after 9h at 1400°C (BSE, 15kV).

Fig. 2d .

 2d Fig. 2d. EDX analysis performed on a remaining tin dioxide grain (white contrast on micrographs).

Fig. 3a .

 3a Fig. 3a. Tin solubility at 1200°C as a function of time (1, 2, 4 and 9 hours) in NC3S and NC6S.

Fig. 4b .

 4b Fig. 4b. Plot of the ratio of Na2O at.% to SiO2 at.% measured in NC6S by EPMA, as a function of time at 1200°C and 1400°C. Theoretical ratio in dotted line.

Fig. 5 .

 5 Fig. 5. Plot of ln[xSn] (dissolved in glass) versus 1/T for the NC3S, NC4S and NC5S glass compositions after heating at 1200°C, 1250°C, 1300°C and 1400°C for 2 hours.

Fig. 6a .

 6a Fig. 6a. Plot of ln[xSn] (dissolved in glass) versus 1/T for NC6S after heating at 1200°C, 1250°C, 1300°C, 1325°Cn 1350°C, 1375°C and 1400°C for 2 hours

Fig. 8 .

 8 Fig. 8. Dissolution and speciation of tin oxide in ternary silicate melts

Fig. 1 .

 1 Fig. 1. Process used for the preparation of the samples for the study of the solubility of tin oxide in silicate melts.

Fig. 2b .

 2b Fig. 2b. Micrograph of NC5S after 2h at 1300°C (BSE, 15kV).

Fig. 2c .

 2c Fig. 2c. Micrograph of NC5S after 9h at 1400°C (BSE, 15kV).

Fig. 2d .

 2d Fig. 2d. EDX analysis performed on a remaining tin dioxide grain (white contrast on micrographs).

Fig. 3a .Fig. 5 .

 3a5 Fig. 3a. Tin solubility at 1200°C as a function of time (1, 2, 4 and 9 hours) in NC3S and NC6S.

Fig. 6a .

 6a Fig. 6a. Plot of ln[xSn] (dissolved in glass) versus 1/T for NC6S after heating at 1200°C, 1250°C, 1300°C, 1325°Cn 1350°C, 1375°C and 1400°C for 2 hours

Fig. 8 .

 8 Fig. 8. Dissolution and speciation of tin oxide in ternary silicate melts

Table 1 .

 1 Ternary glasses compositions. x' is the atomic ratio 2at.%Ca/at.%Na ; x is the average value of the atomic ratios at.%Si/at.%Ca and 2at.%Si/at.%Na. The optical basicities calculated from Eq. (4) are also reported.

		mol.% theoretical
	NCxS Na2O	CaO	SiO2	th	at.%Na(EPMA)	at.%Ca(EPMA)	at.%Si(EPMA)	x'	x
	NC3S	20	20	60	0.718	14.23			6.96		21.57	0.98	3.06
	NC4S	16,7	16,7	66,7	0.679	11.78			6.60		23.04	1.12	3.70
	NC5S	14,3	14,3	71,4	0.650	10.09			5.09		24.89	1.01	4.91
	NC6S	12,5	12,5	75	0.629	8.70			4.31		26.10	0.99	6.02
		Samples		Limit of solubility (at.% of Sn)			Ea (kJ.mol -1 )
				1200°C	1250°C	1300°C	1400°C
			1h	1.38 (0.04)	1.45 (0.07)	1.62 (0.02)	1.99 (0.04)
			2h	1.35 (0.03)	1.44 (0.03)	1.61 (0.10)	1.95 (0.03)
		NC3S									38.5 (2h)
			4h	1.32 (0.01)	1.44 (0.04)	1.58 (0.02)	1.92 (0.03)
			9h	1.30 (0.03)	1.43 (0.03)	1.57 (0.10)	1.90 (0.03)
			1h	1.37 (0.08)	1.49 (0.07)	1.67 (0.05)	2.00 (0.06)
			2h	1.43 (0.02)	1.55 (0.03)	1.67 (0.02)	1.96 (0.03)
		NC4S									32.3 (2h)
			4h	1.47 (0.05)	1.58 (0.03)	1.66 (0.04)	1.96 (0.09)
			9h	1.50 (0.05)	1.59 (0.03)	1.67 (0.02)	1.98 (0.03)
			1h	1.42 (0.10)	1.58 (0.04)	1.72 (0.03)	2.02 (0.06)
			2h	1.53 (0.06)	1.65 (0.05)	1.76 (0.19)	2.08 (0.08)
		NC5S									31.8 (2h)
			4h	1.52 (0.06)	1.60 (0.06)	1.70 (0.05)	1.99 (0.03)
			9h	1.50 (0.03)	1.62 (0.02)	1.68 (0.19)	2.00 (0.03)
			1h	1.06 (0.10)	1.60 (0.04)	1.68 (0.03)	2.06 (0.06)
		NC6S	2h 4h	1.58 (0.05) 1.60 (0.06)	1.64 (0.10) 1.68 (0.06)	1.72 (0.07) 1.90 (0.05)	2.08 (0.06) 2.06 (0.03)	28.2 (2h) 27.9 (9h)
			9h	1.56 (0.03)	1.65 (0.02)	1.82 (0.03)	2.04 (0.03)

Table 2 .

 2 Limit of solubility of tin in the four ternary glasses at four different temperatures and four different times of heat treatment. The activation energies determined from the Arrhenius plots (Fig.5 and 6) are also reported.

Acknowledgement

The authors gratefully acknowledge S. Mathieu and O. Rouer for performing electron-probe micro analysis ("Service Commun de Microscopie Electronique et de Micro-analyse" of the Université of Nancy).


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Emmanuel Guilmeau

Promising thermoelectric properties in Ag x Mo 9 Se 11 compounds "3.4Յ x Յ 3.9…

Polycrystalline Mo 9 cluster chalcogenides Ag x Mo 9 Se 11 ͑3.4Յ x Յ 3.9͒ have been prepared by powder metallurgy techniques, sintered by spark plasma sintering and characterized by x-ray diffraction. Their thermoelectric properties ͑electrical resistivity, thermopower, thermal conductivity͒ have been determined in the 300-800 K temperature range. The Ag x Mo 9 Se 11 compounds show p-type conduction characteristics. The outstanding low lattice thermal conductivities give rise to a rather high value of the dimensionless thermoelectric figure of merit ZT of ϳ0.65 at 800 K for x = 3.8-3.9, making this family of materials particularly promising for thermoelectric power generation applications. © 2011 American Institute of Physics. ͓doi:10.1063/1.3579261͔ Currently, there is a resurgence of interest in thermoelectric materials with enhanced efficiency. These materials can convert directly heat ͑solar heat, geothermal heat, and waste heat from cars or industrial processes, etc.͒ to electricity, and also can act as solid state heat pumps when supplied with electrical current. The conversion efficiency is determined by the dimensionless figure of merit ZT, given by ZT = ͑S 2 / ͒T, where S is the thermopower, the electrical resistivity, the thermal conductivity, and T the absolute temperature at which the properties are measured. [START_REF]CRC Handbook of Thermoelectrics[END_REF] For many years, the best thermoelectric materials, mainly heavily doped semiconductor compounds and alloys such as Bi 2 Te 3 , PbTe, and Si-Ge, possessed ZT values not exceeding unity. [START_REF]CRC Handbook of Thermoelectrics[END_REF][START_REF] Dauscher | Recent Research Developments in Materials Science[END_REF][START_REF] Sootsman | [END_REF] Recent approaches using the "phonon glass electron crystal ͑PGEC͒" concept and more recently the introduction of nanometer-size grains or other nanostructures in a matrix phase have demonstrated the possibility to obtain more performant thermoelectric materials. [START_REF]CRC Handbook of Thermoelectrics[END_REF][START_REF] Dauscher | Recent Research Developments in Materials Science[END_REF][START_REF] Sootsman | [END_REF][4] Several classes of materials have been proofed to epitomize the PGEC concept, specifically partially filled-skutterudites, [START_REF]CRC Handbook of Thermoelectrics[END_REF][START_REF] Dauscher | Recent Research Developments in Materials Science[END_REF][START_REF] Sootsman | [END_REF]5,6 Zn 4 Sb 3 -based materials, [START_REF]CRC Handbook of Thermoelectrics[END_REF][START_REF] Dauscher | Recent Research Developments in Materials Science[END_REF][START_REF] Sootsman | [END_REF]7 clathrates, [START_REF]CRC Handbook of Thermoelectrics[END_REF][START_REF] Dauscher | Recent Research Developments in Materials Science[END_REF][START_REF] Sootsman | [END_REF]8 and semiconducting Chevrel phases. [START_REF] Dauscher | Recent Research Developments in Materials Science[END_REF]9,10 Most Chevrel compounds can be described by the chemical formula MMo 6 X 8 , where M stands for alkaline, alkaline earth, rare earth, actinide elements, or transition metals, and X is usually a chalcogen ͑S, Se, Te͒. [START_REF] Chevrel | Superconductivity in Ternary Compounds I[END_REF] The fundamental structural unit Mo 6 X 8 is the octahedral cluster Mo 6 surrounded by eight chalcogens forming a distorted cube. Mo 6 X 8 units build up a progression of channels, where a variety of M atoms can be accommodated. The best Chevrel phase thermoelectric material discovered to date is Cu/ FeMo 6 Se 8 . 9 This compound has a ZT value near 0.6 at 1150 K. 9 Molybdenum clusters with a nuclearity higher than six can also be obtained and derive from the one-dimensional transface sharing of Mo 6 octahedra. [12][13][14] Among them the ternary Ag x Mo 9 Se 11 ͑with x ϳ 3.6-3.8͒ compounds discovered by Gougeon et al. 12 could be also interesting thermoelectric candidates. Actually, their Fermi level is close to a band edge and their crystal structure is complex ͑high molecular weight, nondirectional bonding, large number of atoms per molecule͒. All these conditions are beneficial to achieve a low thermal conductivity at high temperature. So far, the electrical resistivity at low temperature has only been investigated. 12 In this letter we report on the high temperature thermoelectric properties of Ag x Mo 9 Se 11 ͑3.4Յ x Յ 3.9͒ ternary compounds and show that the ZT values surpass those of the best Chevrel phase.

Stoichiometric amounts of highly pure Ag, Mo, and MoSe 2 powders were used for the preparation of the materials by solid state reaction, already described. 13 The obtained samples were ground and subsequently densified by spark plasma sintering ͑SPS͒ at 1050 °C with a holding time of 10 min under 80 MPa in an argon atmosphere. X-ray diffraction ͑XRD͒ analyses were performed on a monochromatized Cu K␣1 Bruker D8 Advance diffractometer. Rietveld refinement was realized using FULLPROF software. 15 The temperature dependences of the thermopower and electrical resistivity were measured simultaneously in the 300-800 K temperature range by a standard four-probe method using a ULVAC-ZEM3 device on bar-shaped samples, which were cut perpendicularly to the press direction. The thermal diffusivity ␣ was measured in the same temperature range by the laser flash method ͑NETZSCH, LFA 427͒ on disk-shaped samples, which were cut perpendicularly to the press direction. The specific heat C p was determined using a differential scanning calorimetry ͑DSC͒ apparatus ͑NETZSCH, DSC 403 F3͒. The thermal conductivity was evaluated through the standard equation = ␣ ducted at room temperature using a five probes method on a Quantum Design model physical property measurement system. Sound velocity measurements were made at a frequency of 10 MHz using the ultrasonic pulse-echo method. The mean sound velocity v was determined from v = ͓1 / ͑3v l 3 ͒ +2/ ͑3v s 3 ͔͒ -1/3 , where v l and v s are the longitudinal and the shear wave velocities, respectively.

The Ag x Mo 9 Se 11 compounds crystallize in the orthorhombic space group Cmcm. The building block is the Mo 9 Se 11 Se 6 cluster unit, as shown in Fig. 1͑a͒. The bioctahedral Mo 9 cluster results from the uniaxial face-sharing condensation of two Mo 6 Se 8 units with loss of Se atoms belonging to the shared face. This cagelike structure can accommodate Ag atoms, which are delocalized over four crystallographically independent sites. Pioneer works on Ag x Mo 9 Se 11 single crystals prepared by chemical transport in a vapor phase led to x values limited to 3.6-3.76. [START_REF] Chevrel | Superconductivity in Ternary Compounds I[END_REF] In this letter, single phase polycrystalline samples have been obtained for 3.4Յ x Յ 3.9. The presence of secondary phases, mainly MoSe 2 and the Chevrel phase Ag y Mo 6 Se 8 or Ag metal, has been detected outside this range of content. A typical XRD pattern is displayed in Fig. 1͑b͒ for Ag 3.7 Mo 9 Se 11 . The enhancement of the unit cell volume with the Ag filling level is also shown in the inset.

The electrical resistivity and thermopower of Ag x Mo 9 Se 11 are displayed as a function of temperature in Fig. 2. The resistivity shows a linear increase in all the temperature range studied, whatever the sample is, characteristic of a metal-like behavior. For a given temperature, increases with x, e.g., from around 32 ⍀ m for x = 3.4 to 64 ⍀ m for x = 3.9 at 800 K. The positive sign of S indicates that holes are the majority carriers in the system. The S values increase with both temperature and Ag concentration, ranging from 140 V / K for x = 3.4 to 188 V / K for x = 3.9 at 800 K. For the analogous Chevrel phases, one of the most important electronic parameter is the metallic electron count ͑MEC͒, also termed as valence electron count ͑VEC͒, i.e., the total number of valence electrons per Mo 6 cluster. MEC is a convenient measure for determining the degree of filling of the valence band near the Fermi level. [START_REF] Yvon | Current Topics in Materials Science[END_REF] Chevrel phases are reported with MEC values between 20 and 24, this last value turning the material to a semiconducting state. The compositions with a MEC lower than 24 are metallic and this has been experimentally confirmed in binary Chevrel compounds inserted with Cu, Cu/Fe, or Ag atoms. 9,[START_REF] Shi | [END_REF] High MEC values can be obtained either by increasing the concentration of filler atoms or by substitutions in the framework of Mo 6 Se 8 . For example, mixed clusters such as Mo 4 Ru 2 Se 8 or Mo 2 Re 4 Se 8 and the Ti filled Mo 6 Se 8 are found to be semiconductors. 9,[18][19][20] A previous theoretical study on the Mo 9 clusters has shown that the MEC in the metastable binary Mo 9 Se 11 is 32 and that the semiconducting state can be reached for 36. 21,22 Supposing Ag to be formally +1, increasing x in Ag x Mo 9 Se 11 should then result in a decrease in the hole concentration and hence in an increase in both the resistivity and the thermopower. Our results support well this picture. However, the galvanomagnetic study indicates that the electronic properties of these interesting compounds cannot be classified as usual. Actually, Hall coefficients of Ag x Mo 9 Se 11 ͑3.4Յ x Յ 3.9͒ at 300 K are negative and field independent from 10 to 50 kOe. This field independence suggests that the weak field limit always remains. This is expected if the states near the Fermi energy have significant d character, leading to high effective masses and correspondingly low mobilities. Given that S is positive at 300 K, it is clear that both holes and electrons are contributing to the conduction.

The temperature dependence of the thermal conductivity of the Ag x Mo 9 Se 11 compounds is plotted in Fig. 2͑c͒, and compared to that of Cu/ FeMo 6 Se 8 . 9 One can see that whatever x is, in Ag x Mo 9 Se 11 compounds is lower than in the Chevrel phase. At 300 K, the values range from 0.8 to 1.2 W/mK for x = 3.9 and x = 3.4, respectively. The value remains nearly constant with temperature, indicating that the thermal conductivity approaches a glasslike limit. It is worthy to point out here that was not measured along the same direction as and S. However, by measuring the thermal conductivity along and perpendicular to the press direction for a given composition, no appreciable texture influence could be detected ͓Fig. 2͑d͔͒. Using the measured values of and supposing that the Wiedemann-Franz law is valid, the lattice thermal conductivity, L , was evaluated. The values, displayed in Fig. 2͑e͒, are in the range 0.4-0.7 W/mK over the entire range studied. These values are lower than those reported for the best thermoelectric materials such as partially filled skutterudites. 23 Figure 2͑e͒ also shows an estimate of the "minimum" lattice thermal conductivity of Ag x Mo 9 Se 11 , when the mean free path l reaches the interatomic distance. We used, for this estimate, the kinetic formula L =1/ 3C v vl, where the specific heat C v was taken equal to C p , the determined sound velocity v is 2110 m/s and the interatomic distance l is 0.3 nm. We can note that L of the studied compounds approaches the minimum value possible. The origin of the very low lattice thermal conductivity in these compounds is not yet fully addressed. We rule out any influence of the microstructure because scanning electron microscopy images indicate an average grain size of about some micrometers. Rather, we speculate that the crystal complexity as well as a static and/or dynamic disorder coming from the Ag atoms should play an important role.

The dimensionless figure of merit ZT as a function of temperature for all the compounds is displayed in Fig. 2͑f͒ for all the compounds together with the Cu/ FeMo 6 Se 8 . 9 The Ag x Mo 9 Se 11 ternary compounds possess ZT values greater than the best Chevrel phase over the temperature range investigated. ZT reaches the value of 0.65 at 800 K in compounds having the highest Ag nominal content ͑x = 3.8-3.9͒.

These encouraging results show the interest for further investigating this family of Mo 9 chalcogenide compounds as potential thermoelectric materials. It is of particular importance to determine the origin behind such surprisingly low lattice thermal conductivity. The relevant study is currently underway.

͑Received 6

 6 January 2011; accepted 4 March 2011; published online 21 April 2011͒

FIG. 1 .

 1 FIG. 1. ͑Color online͒ ͑a͒ Crystalline structure highlighting the Mo 9 Se 11 Se 6 cluster unit. ͑b͒ XRD pattern for Ag 3.7 Mo 9 Se 11 before SPS sintering ͑similar pattern after SPS sintering͒. The unit cell volume as a function of Ag content x in Ag x Mo 9 Se 11 is shown in the inset.

The authors are thankful to Professor D. P. Almond in University of Bath ͑U.K.͒ for the sound velocity measurement. This work was supported by Region Lorraine and INPL. M.C. greatly acknowledges the financial support of ESA ͑ESA/ESTEC Contract 22955/09/NL/PA͒.


Pierre Aboulker

Guillaume Aubian

Pierre Charbit

Decomposing and colouring some locally semicomplete digraphs

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and the in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decomposition theorems. As a consequence we obtain several applications, among which an answer to a conjecture of Naserasr and the first and third authors: if an oriented graph is such that the out-neighbourhood of every vertex induces a transitive tournament, then one can partition its vertex set into two acyclic digraphs.

Introduction

In this paper all directed graphs (digraphs in short) are simple, i.e. contain no loop and no multi-arc. If in addition a digraph contains no digon (a cycle on two vertices), we say that it is an oriented graph.

For a vertex x of a digraph D, we denote by x + (D) (resp. x -(D)) the set of its out-neighbours (resp. in-neighbours) :

x + (D) = {y ∈ V (D), xy ∈ A(D)} x -(D) = {y ∈ V (D), yx ∈ A(D)}
If there is no ambiguity on the digraph, we will simply use x + and x -.

Given a digraph D = (V, A), its underlying graph is the unoriented graph G = (V, E) such that xy ∈ E if xy ∈ A or yx ∈ A. We say that a digraph is connected if its underlying graph is connected as an unoriented graph. A semicomplete digraph is a digraph whose underlying graph is a complete graph. A tournament is a semicomplete oriented graph. A digraph D is strongly connected if there is a directed path between each ordered pair of vertices. In this case we say that D is a strong digraph.

We denote by ← → K 2 the digraph on two vertices with an arc in both directions, by C k the directed cycle on k vertices and by T T k the unique acyclic tournament on n vertices, called transitive tournament. The oriented graph on three vertices with a vertex of out-degree 2 (resp of in-degree 2) and two vertices of in-degree 1 (resp. two vertices of out-degree 1) is called S + 2 (resp. S - 2 ). For a class P of digraphs (like semicomplete, tournament, acyclic), a digraph is locally out-P (resp. locally in-P) if for every vertex x, x + (resp. x -) induces a digraph in P . For example, a digraph D is locally out-semicomplete if the out-neighborhood of every vertex of D induces a semicomplete digraph. Finally, we will say that a digraph is locally P if it is both locally out-P and locally in-P. We make one exception for one of the main class of this paper: for the oriented graphs for which the out-neighborhood of every vertex is a transitive tournament, we will use the term "out-transitive oriented graphs" instead of the heavier and possibly confusing "out-transitive tournament oriented graphs".

A linear order on a digraph D is an order O := (v 1 , v 2 , . . . , v n ) of its vertices. Two orders O 1 and O 2 are equivalent if O 1 := (v 1 , v 2 , . . . , v n ) and O 2 = (v k , v k+1 , . . . , v n , v 1 , v 2 , . . . , v k-1 ) for some k. An equivalence class for this relation is called a cyclic order of D. For two vertices v i and v j in a linear order the cyclic interval [v i , v j ] is defined as follow:

[v i , v j ] = {v k , k ∈ [i, j]} if i < j {v k , k ∈]j, i[} if i ≥ j
Note that cyclic intervals only depend on the cyclic order and not on a linear order chosen as a representative. As usual,

]v i , v j [= [v i , v j ] \ {v i , v j }.
Semicomplete digraphs are well studied and a natural and fruitful way to extend results on this class is to look at the class of locally semicomplete digraphs. Introduced in 1990 by Bang-Jensen [START_REF] Bang-Jensen | Locally semicomplete digraphs: a generalization of tournaments[END_REF], locally semicomplete digraphs have since then been the topic of more than 100 research papers, and a whole chapter in [START_REF] Bang-Jensen | Classes of Directed Graphs[END_REF] is devoted to this class. A particularly nice result in this area is one of Huang that gives a geometric characterization of locally transitive digraphs. We state it here in the particular case of oriented graphs.

Theorem 1.1 (Huang,[START_REF] Huang | Tournament-like Oriented Graphs[END_REF])

If D = (V, A
) is a connected oriented graph, then the two conditions below are equivalent 1. for every vertex x, both x + and x -induce a transitive tournament.

2. there exists a cyclic order of the vertices of D such that ∀xy ∈ A, ∀z ∈]x, y[, xz ∈ A and zy ∈ A Any oriented graph (strong or not) that satisfies the second condition above is called a round oriented graph. In other words, for every vertex x, x + (resp. x -) consists in a cyclic interval placed just after (resp. before) x in the cyclic order. Note that a round oriented graph is strong if and only if every vertex has at least one in-neighbour. This is because the cyclic order given by the theorem then yields an Hamiltonian cycle. By the similar observation, if a round oriented graph is not strong, then it is in fact acyclic.

Our first result is a generalization of the theorem above in the particular case of strong oriented graphs.

Theorem 2.1

Let D be a strong oriented graph. Then conditions below are equivalent.

1. for every vertex x, x + induces a tournament and x -induces an acyclic digraph 2. there exists a cyclic order of the vertices of D such that ∀xy ∈ A, ∀z ∈]x, y[, zy ∈ A Again condition 2 can be seen as the property that for every vertex x, x -consists in a cyclic interval placed just before x in the order (see Figure 1). Following the terminology of [START_REF] Bang-Jensen | Classes of Directed Graphs[END_REF], any oriented graphs satisfying condition 2 of the theorem above will be called in-round.

In [START_REF] Li | A sufficient condition for a digraph to be positive round[END_REF], we note that the authors prove a similar result giving an alternate condition 1: they ask that for every vertex x, x + induces a transitive tournament and x -induces an acyclic digraph with a hamiltonian path. These additional conditions (which are easily seen in fact to be implied by condition 2) are unnecessary, which makes our theorem slightly stronger. Moreover, our proof, exposed in section 2, is also much shorter.

Figure 1: An in-round oriented graph that is not round In fact, Theorem 2.1 is the first step towards our main structural result, which is a decomposition theorem for the class of strong locally out-transitive digraphs. If H is a subdigraph of D, we define the contraction D/H as the digraph obtained by removing all vertices of H, then adding a new vertex h such that xh (resp. hx) is an arc of D/H if x + ∩ H (resp x -∩ H) is non empty. Beware that in general D/H might contain digons (even if D does not), eventhough in our case this never happens.

Theorem 2.3

If D is a strong locally out-transitive oriented graph, then there exists a partition of its set of vertices into strong subdigraphs D 1 , . . . , D k such that the digraph obtained by contraction of every D i is a strong in-round oriented graph.

We give two applications of this theorem. The first one is an acyclic colouring result: we prove that if D is an out-transitive oriented graph, then it can be partitioned into two acyclic induce subdigraphs. This proves a conjecture of [START_REF] Aboulker | Extension of the Gyárfás-Sumner conjecture to digraphs submitted[END_REF]. Proof together with context of the conjecture is given in Section 3.1. This result was independently proof by Raphael Steiner, see Remark 1.2. The second one is a proof that locally in-tournaments satisfy the famous Caccetta-Häggkvist conjecture. This result is mentioned in [START_REF] Razborov | On the Caccetta-Häggkvist Conjecture with Forbidden Subgraphs[END_REF] as an unpublished but not easy result of Paul Seymour but here the idea is to show how our decomposition theorem more or less directly implies the result. The proof is given in Section 3.2

Finally, in Section 4, we use the techniques developed for the proof of Theorem 3.3 to prove a structural theorem (see Theorem 4.2) for the class of locally-semicomplete digraphs. Such a theorem is given in [START_REF] Bang-Jensen | A classification of locally semicomplete digraphs Discrete[END_REF] (see Theorem 3.12), but we feel that our theorem is simpler and might have some interesting applications. We propose some illustrations after the proof of the theorem.

One idea behind the paper is to promote the idea of finding decomposition theorem for classes of digraphs, that is a theorem whose statement is of the kind: either a graph in this class is "basic" (belongs to some simple subclass), or it can be decomposed in some prescribed ways. These kinds of decomposition theorems proved to be a very powerful strategy in the world of undirected graphs (the most famous example being the celebrated proof of the perfect graph conjecture by Chudnovsky et al in [START_REF] Chudnovsky | The Strong Perfect Graph Theorem[END_REF]), and there are to our knowledge not so many theorems of this kind in the world of directed graphs, and there is no reason to believe that it could not be as effective in this setting.

Remark 1.2. A week prior to the submission of this paper, R. Steiner published on arXiv a very nice paper [START_REF] Steiner | On coloring digraphs with forbidden induced subgraphs[END_REF] containing another proof of the colouring result of locally out-transitive oriented graphs mentioned above (as well as other results about acyclic colourings). Even though some of the ingredients are in common, the proof is longer and is different as it is an entirely inductive proof whereas ours relies on the structure theorem mentioned above.

Let D be a strong oriented graph. Then conditions below are equivalent.

1. for every vertex x, x + induces a tournament and x -induces an acyclic digraph 2. there exists a cyclic order of the vertices of D such that

∀xy ∈ A, ∀z ∈]x, y[, zy ∈ A
We recall that we will use the term in-round for any oriented graph satisfying condition 2.

Proof of Theorem 2.1 : The easy direction is 2 implies 1. Indeed, let x, y, z be such that y, z ⊂ x + and assume w.l.o.g that z ∈ [x, y]. Then by 2 we have that zy ∈ A, so x + is a tournament. Assume now that x -contains a directed cycle C, and let y be the vertex of C such that C \ y ⊂ [y, x] (the left most vertex of C in the representative of the cyclic order which ends in x). Let z be the predecessor of y on C. Now we have that x ∈ [zy] and so by 2 there must be an arc xy, which contradicts the fact that y ∈ x -(there are no digon here). Now assume 1. For every vertex x, x -induces a non empty acyclic oriented graph, and hence contains a vertex y such that y + ∩ x -= ∅ (take the last vertex in a topological ordering of x -). For every x we arbitrarily choose one such vertex and denote it by f (x). If z is an out-neighbour of f (x), then since f (x) + induces a tournament, z and x must be connected by an arc, and this cannot be zx by definition of f (x), so there must be an arc xz. We have therefore f (x) + \ {x} ⊂ x + for all x.

Now let H be the graph induced by the arcs f (x)x. Each vertex of H has in-degree exactly 1, so H contains a cycle C. If C does not span all vertices of D, then since D is strong, there exists an arc xy in D where x ∈ C and y ∈ C. Because of the property above the whole cycle C must be contained in y -, which contradicts 1. So H consists in an Hamiltonian cycle and the property f (x) + ⊂ x + exactly translates into 2 for the cyclic order defined by this Hamiltonian cycle.

Remark 2.2. We point out that the proof above shows that Property 2 implies in fact that every outneighbourhood in a in-round oriented graph induces a transitive tournament (even if it is not necessary to add it in Property 1 to get the equivalence). Thus in-round strong oriented graphs can also be seen as strong oriented graphs that are locally out-transitive and locally in-acyclic as proved proved in [START_REF] Li | A sufficient condition for a digraph to be positive round[END_REF].

We now recall the statement of the main structural theorem mentioned in the introduction.

Theorem 2.3

If D is a strong locally out-transitive oriented graph, then there exists a partition of its set of vertices into strong subdigraphs D 1 , . . . , D k such that the digraph obtained by contraction of every D i is an strong in-round oriented graph.

Let us start with a lemma about locally out-semicomplete digraphs (be aware that the rest of this section is about oriented graphs and not digraphs, but we state it here for digraphs as we will make use of it in Section 4, where we study digraphs). Proof : Let h and h be vertices of H such that hz and hh are arcs of D. Because D is out-semicomplete and z + ∩ H = ∅, h z must be an arc. Since H is strong, we get that h -∩ H = H.

Proof of Theorem 2.3 : From now on D will denote a strong locally out-transitive oriented graph. We define a hub to be a subset of vertices H of D such that

• H induces a strong oriented graph,

• there exists x / ∈ H such that H ⊆ x -.

Note that a hub is necessarily a strict subset of V (D). A hub is trivial if it is reduced to one vertex.

Recall that an oriented graph is in-round if the in-neighborhood of each vertex induces an acyclic oriented graph, and the out-neighborhood induces a tournament. Hence, since a cycle that is out-dominated is a hub, D is in-round if and only if there is no non trivial hub.

Assume D is not in-round, and consider a hub H that is (inclusion-wise) maximal. We want to prove that D/H is a locally out-transitive oriented graph. Assume first by contradiction that there exists a vertex z / ∈ H that is mixed for H, that is such that there exists {h, h } ⊂ H with (hz, zh ) ∈ A(D) 2 . Since H is strong, there is a directed path from h to h . Due to the fact that h + induces a tournament, by considering such h and h' with a path of minimal length, one can assume that hh is an arc. There exists x in D such that H ⊂ x -, and because D does not contain S + 2 there must be an arc between x and z. It cannot be xz because in that case h + would contain the directed triangle xzh which is not possible so zx must be an arc. But then H ∪ {z} is a hub that contradicts the maximality of H.

So using Lemma 2.4, we observe that eventually there are three types of vertices outside H: the one that have no arc to or from H, the one that have out-neighbours in H but no in-neighbour, and the ones that are out-neighbour of every vertex of H. One can now easily check that D/H is locally out-transitive.

Consider now two distinct maximal hubs H1 and H2. We want to prove that their intersection is empty. For i = 1, 2, let h1 be a vertex that in-dominates Hi. Note that it is not possible that h1 ∈ H2 and h2 ∈ H1 simultaneously, as this would imply a digon between the two vertices. So without loss of generality, assume h1 / ∈ H2. But now if H1 ∩ H2 is non empty, then h1 has a in neighbour in H2 and by Lemma 2.4 and since we proved that no vertex can be mixed for H2, we have that h1 in-dominates and H1 ∪ H2, which is therefore also a hub (it is strongly connected because the intersection is non empty). This contradicts the maximality of H1 and H2

Eventually we have proven that maximal hubs define a partition of the vertex set (it is trivial that every singleton is a hub). Moreover, by what precedes, if there is an arc xy from a maximal hub H1 to a maximal hub H2, then y is an out-neighbour of every vertex in H1. We summarize this with the following claim (the transitive tournament fact is due to the graph being locally out-transitive). Claim 2.5. Maximal hubs form a partition of the vertex set. Moreover, if H1 and H2 are two maximal hubs, then between them:

• either there is no arc,

• either all arcs from H1 to a subset of H2 inducing a transitive tournament and no other arc,

• either all arcs from H2 to a subset of H1 inducing a transitive tournament and no other arc.

Let D be the digraph obtained by contracting every maximal hub. If D contains a non trivial hub H , then by the fact above it is clear that the set H of vertices of D that are mapped to vertices in H by the contraction form a hub that would contradict the maximality of the hubs that were contracted to obtain D . So D contains no non trivial hub, so no dominated cycle, and hence by Theorem 2.1, it is in-round. This concludes the proof of Theorem 2.3.

3 Applications of Theorem 2.3

Acyclic Colourings

An acyclic colouring of a digraph is an assignment of colours to the vertices such that each colour class induces an acyclic subdigraph, that is a subdigraph containing no directed cycle. The acyclic chromatic number, or simply dichromatic number, of a digraph D, is defined to be the smallest number of colours required for an acyclic colouring of D and is dentoed -→ χ (D). This notion was first introduced in 1982 by

Neumann-Lara [START_REF] Neumann-Lara | The dichromatic number of a digraph[END_REF] and has attracted a lot of attention in the past decade (see for example [START_REF] Aboulker | Thomassé Subdivisions in digraphs of large out-degree or large dichromatic number[END_REF][START_REF] Bensmail | List colouring digraphs[END_REF][START_REF] Hoshino | Kawarabayashi The edge density of critical digraphs[END_REF][START_REF]Hochstättler A flow theory for the dichromatic number[END_REF][START_REF] Kostochka | Stiebitz The Minimum Number of Edges in 4-Critical Digraphs of Given Order[END_REF][START_REF] Li | Planar digraphs of digirth four are 2-colourable[END_REF][START_REF] Gishboliner | Dichromatic number and forced subdivisions[END_REF][START_REF] Steiner | A note on coloring digraphs of large girth[END_REF]) as it seems to be the natural generalization for digraphs of the usual chromatic number.

A hero is a tournament H such that every tournament not containing H have bounded dichromatic number. In [START_REF] Berger | Tournaments and colouring[END_REF], Berger et al give a full description of heros. Extending the notion of hero, a set F of digraphs is said to be heroic if every digraph not containing any member of F as an induced subdigraph has bounded dichromatic number. In [START_REF] Aboulker | Extension of the Gyárfás-Sumner conjecture to digraphs submitted[END_REF], the first and third authors together with Naserasr proposed the following conjecture, along with a proof of the only if part. It can be seen as an analogue of the well-known Gyárfás-Sumner Conjecture [START_REF] Gyárfás | Problems from the world surrounding perfect graphs[END_REF][START_REF] Sumner | Subtrees of a graph and the chromatic number. The theory and applications of graphs[END_REF] for oriented graphs.

Conjecture 3.1 ([2])

Let H be a hero and let F be an oriented forest. The set { ← → K 2 , H, F } is heroic if and only if:

• either F is the disjoint union of oriented stars,

• or H is a transitive tournament.

This conjecture is still widely open. The first case that was left in [START_REF] Aboulker | Extension of the Gyárfás-Sumner conjecture to digraphs submitted[END_REF] is the case F = S + 2 , and H = C 3 and it was conjectured that digraphs without any induced subgraph in { ← → K 2 , C 3 , S + 2 } have dichromatic number at most two. As mentioned in the introduction, we now prove this result, and in fact a stronger result. We recall that in [START_REF] Steiner | On coloring digraphs with forbidden induced subgraphs[END_REF], Raphael Steiner independently proved the same result as well as other special cases of Conjecture 3.1, see Remark 1.2.

Theorem 3.2

Every locally out-transitive oriented graph has dichromatic number at most 2.

Note that it indeed extends the question mentioned above as it amounts to forbidding ← → K 2 , S + 2 and the tournament on 4 vertices built by taking a directed triangle C 3 and adding a vertex with an arc going to the three other vertices. As already said, forbidding S + 2 implies that every out-neighbourhood induces a tournament, and forbidding this 4-vertex tournament implies that every out neighbourhood must induce a C 3 -free tournament, hence acyclic (every tournament containing a directed cycle must contain a C 3 ).

To ease the induction proof we will prove the following stronger result.

Theorem 3.3

Let D be a locally out-transitive oriented graph and T a subdigraph of D inducing a transitive tournament. Then D admits a proper 2-dicolouring such that all vertices of T receive the same colour.

The first step towards the proof of this theorem is to prove the exact same statement for the class of in-round oriented graphs (which we recall can also be seen as locally out-transitive and locally in-acyclic oriented graphs).

Proposition 3.4

Every in-round oriented graph has dichromatic number at most 2. More precisely, for every vertex x, there exists a valid 2-dicolouring such that {x} ∪ x + is monochromatic.

Proof : We only need to prove it when the oriented graph is strong since a 2-colouring of each strong component yields a valid 2-colouring of the whole oriented graph. Consider the cyclic order given by the definition of in-round and pick any vertex x. Let y be the vertex such that xy is a longest arc, that is the arc such that the interval [x, y] contains the maximum number of vertices. This implies that in the linear order given by the interval ]y, x[ all arcs go forward since a back arc x y would force the arc xy contradicting the maximality of xy. So ]y, x[ induces an acyclic oriented graph. Moreover [x, y] induces an acyclic oriented graph since it is included in y -and by definition of the in-round cyclic order. This concludes the proof.

We are now ready to prove Theorem 3.3.

Proof of Theorem 3.3 : Again we can assume that our digraph D is strong, and we proceed by induction on the number of vertices of D. Let T be a transitive tournament in D. We consider the decomposition into maximal hubs given by Theorem 2.3 and label the hubs H1, . . . , H k so that the order corresponds to the cyclic order h1, . . . , h k of the contracted in-round oriented D . Since the theorem guarantees that D contains no digons, there can exist arcs in only one direction between two distinct Hi. Moreover, Lemma 2.4 implies that if there is an arc xy from Hi to Hj, then there is an arc from every vertex in Hi to y and by the in-round property of D that the same is true for any vertex in H k for k ∈ [i, j[. We define Ti for i = 1, . . . , k to be the set of vertices in Hi that have an in-neighbour out of Hi. By the remark just above, it in fact consists in x + ∩ Hi for every x ∈ Hi-1. Note also that if a cycle of D is not included in a maximal hub, then if it intersects Hi, it must intersect Ti. Let s be the source in the transitive tournament T , and without loss of generality assume it belongs to H1. Note that because every other vertex in T is an out-neighbour of s, we have that T ∩ Hi ⊂ Ti for every i ≥ 2. Now we are ready to define our colouring. First we consider the in-round oriented graph D and denote by T the subtournament (transitive) in D consisting in all hi such that T ∩ Hi is non empty. By Proposition 3.4, there exists a valid 2-dicolouring of D such that the out-neighbourhood of h1 (which contains T ) is monochromatic, say coloured 1. Now by induction we can ask for every i ≥ 2 for a colouring of Hi such that all vertices of Ti get the colour of hi in the colouring of D defined above. For i = 1 we ask by induction for a colouring of H1 such that every vertex of T ∩ H1 gets the colour of h1, that is 1.

First, note that in this colouring, every vertex of T gets colour 1 because of the assumption on the colouring of D . We now need to prove that that this colouring is a valid 2-dicolouring of D. Assume by contradiction that C is a monochromatic cycle and consider a minimal such cycle. C is not included in some Hi since the colouring is valid in every hub. So C hits more than one hub, and since the vertices in each Hi have the same out-neighbours out of Hi, the minimality of C implies that C contains at most one vertex from each hub Hi and this vertex must belong to Ti (otherwise it cannot have an in-neighbour out of Hi). The only case where this does not yield directly a monochromatic cycle (and hence a contradiction) in the contracted digraph D is if C is coloured 2 and contains a vertex x in T1 \ T . Let y be its successor on the cycle. Then y belongs to some Tj where j is such that h1hj is an arc of D . But by assumption on the colouring of D this implies that hj gets colour 1 and therefore y must get colour 1, which is a contradiction that finishes the proof of Theorem 3.3.

About Caccetta-Häggkvist Conjecture

A beautiful and famous conjecture due to Caccetta and Häggkvist states states the following.

Conjecture 3.5 (Caccetta-Häggkvist)

Let k ≥ 2 be an integer. Every digraph D on n vertices with no directed circuits of length at most k contains a vertex of out-degree less than n/k.

The case k = 2 is trivial but the case k = 3 is still widely open and has attracted a lot of attention. In [START_REF] Razborov | On the Caccetta-Häggkvist Conjecture with Forbidden Subgraphs[END_REF] (see page 3), it is mentioned that for k = 3, while adding the hypothesis that the graph has no S + 2 makes it very easy, the dual case of forbidding S - 2 was proven by Seymour but is "substantially more difficult". Here we prove that this comes as an easy consequence of Theorem 2.3 and Theorem 2.1, for any value k ≥ 3. Theorem 3.6

Let D be locally in-tournament oriented graph on n vertices with no directed cycle of length at most k. Then D contains a vertex of out-degree less than n/k. Theorem 2.3 and Theorem 2.1 were designed for locally out-tournament but of course by reversing the arcs we get an equivalent statement for locally in-tournament (here an out-round digraph is a digraph obtained by an in-round digraph by reversing all the arcs). We combine both to get this corollary.

Corollary 3.7

If D is a strong locally in-tournament that does not contain any C 3 , there exists a partition of its set of vertices into strong subdigraphs D 1 , . . . , D k such that the digraph D obtained by contraction of every D i is a strong oriented graph which is out-round, that is admits a cyclic order on its vertices such that

∀x, y, z ∈ V (D ) (xy ∈ A ∧ z ∈]x, y[) ⇒ xz ∈ A
Proof of Theorem 3.6 : Let D be digraph as in the statement of the theorem and denote by n the number of its vertices. First observe that we can assume D to be strong, since one can apply it to a terminal strong component. Since k ≥ 3, notice that D does not contain C3 so we can apply Corollary 3.7 above. We begin by proving a wighted version for out-round oriented graphs.

Lemma 3.8. Let D be a strong out-round oriented graph with no cycle C3 and let w be a positive weight function on the vertices of D. Denote by W the sum of all weights. Then there exists a vertex u such that

v∈u + w(v) < W -w(v) k
Let us show first that this lemma indeed implies the theorem. Since D admits an out-round quotient by Corollary 3.7, we can apply the lemma to the quotient with weights being the sizes of the contracted subdigraphs. Let Di be the subdigraph corresponding to the vertex of small out-degree given by the claim. Now by applying induction to Di, one finds a vertex in Di with out-degree (strictly) less than |Di|/k, which combined with the lemma gives the desired result.

Let us now prove the lemma. Consider the cycle order as in Corollary 3.7. For every vertex x, denote by f (x) the last of its out-neighbour along the cyclic order and by φ(x) the quantity y∈x + w(y). Observe that due to the out-round structure, φ(x) is exactly the sum of weights in the interval ]x, f (x)]. We denote also by f (i) (x) = f (f (i-1) (x) with the convention f (0) (x) = x.

Consider a path on k vertices of the form xf (x)f (f (x)) . . . f (k-1) (x). Observe that because of the out-round structure and the non existence of a cycle of length at most k the path does not cycle completely around the cyclic order. Furthermore for the same reason we have that any two intervals ]f (i) (x)f (i+1) (x)] for distinct values i = i are disjoint. Eventually we have that for every vertex

x k-1 i=0 φ (i) (x) ≤ W -w(x)
Now consider some cycle C formed by the arcs xf (x). By summing the above inequality for all vertices in the cycle one gets k

x∈V (C) φ(x) ≤ W |C| - x∈V (C) w(x)
So there must exist a vertex in V (C) that satisfies :

kφ(x) + w(x) ≤ W
which is exactly the assertion of the lemma.

Structure of locally semicomplete digraphs

In this section digons are allowed and we focus on locally semicomplete digraphs which we recall is the class of digraphs such that the in-neighbourhood and out-neighbourhood of any vertex is semicomplete.

A statement in the flavour of Conjecture 3.1 is easy to prove. The proof was independently obtained by Raphael Steiner [START_REF] Steiner | On coloring digraphs with forbidden induced subgraphs[END_REF]. Theorem 4.1

For every hero H, H-free locally tournaments have bounded dichromatic number. More precisely, it is bounded by at most twice the maximum dichromatic number of a H-free tournament.

Proof : Let k be the maximum dichromatic number of an H-free tournament. Let x be any vertex. By induction on the number of vertices, we know that the oriented graph induced by the set of non neighbours N = V \ ({x} ∪ x + ∪ x -) is 2k colourable so we first properly colour N with colours in {1, . . . , 2k}. Now since x + and x -are tournaments, they are k-colourable by hypothesis. We can thus use colours {1, . . . , k} for a proper colouring of x -, and colours {k + 1, . . . , 2k} for a proper colouring of x + , give x any colour, and it is not difficult to check that this gives a valid 2k-colouring of D (the main fact to observe is that there are no arcs from x -to N , and from N to x + ).

The purpose of this section is to use our techniques to give a proof of the following structural theorem for the class of locally semicomplete digraphs. As already mentioned in the introduction, such a theorem is given in [START_REF] Bang-Jensen | A classification of locally semicomplete digraphs Discrete[END_REF] (see Theorem 3.12), but our theorem is much simpler to state and might have some interesting applications. In fact the theorem of [START_REF] Bang-Jensen | A classification of locally semicomplete digraphs Discrete[END_REF] has the same first two cases below, but their third case (called evil in Chapter 6 of the monograph [START_REF] Bang-Jensen | Digraphs. Theory, algorithms and applications[END_REF]) has a more complicated description that seem to us less easy to handle for applications. We propose some illustrations after the proof of the theorem.

We need some vocabulary first. A vertex x out-dominates (resp. in-dominates) a set of vertices X if X ⊆ x + (resp. X ⊆ x -). A vertex x strictly out-dominates (resp. strictly in-dominates) a set of vertices X if X ⊆ x + \ x -(resp. X ⊆ x -\ x + ). A set of vertices X out-dominates (resp. strictly out-dominates, in-dominates, strictly in-dominates) a set of vertices Y if every vertex of X out-dominates (resp. strictly out-dominates, in-dominates, strictly in-dominates) Y . Also, we call universal vertex a vertex x such that

x + = x -= V (D) \ {x}. Theorem 4.2
Let D a connected locally semicomplete digraph, then either :

• D is semicomplete with a universal vertex.

• There exists a partition of V (D) into k ≥ 2 subsets each inducing strong connected semicomplete digraphs such that the digraph obtained by contracting every member of the partition is a round oriented graph.

• there exists a partition of V (D) into four sets E, F , G and H such that :

-F and H are non empty, and one of E and G is non empty.

- -∀x ∈ G, x + ∩ E = ∅ and x -∩ E = ∅ For this proof, we need to relax the notion of hub of the previous section: a set X ⊆ V is called a weak hub if there exists a vertex which strictly in-dominates or strictly out-dominates X. The "strictly" part of this definition was implicit before for hubs since we were in the context of oriented graphs. A weak hub X is said to be mixed if there exist vertices x / ∈ X and u, v ∈ X such that both xu, vx are arcs. We split the proof of Theorem 4.2 into two parts depending on the existence of a maximal weak hub that is mixed. When there are no such subsets, the proof will have a lot of similarities with the one of Theorem 2.3. Lemma 4.3. Let D be a connected locally semicomplete digraph such that no maximal weak hub is mixed. Then either D is semicomplete with a universal vertex, or there exist k ≥ 2 disjoint sets X 1 , X 2 . . . X k of vertices such that D[X i ] is strong for i = 1, . . . , k, and the digraph obtained by contraction of every X i is a round oriented graph.

Proof of Lemma 4.3 : Let D be a connected locally semicomplete digraph and assume it is not semicomplete with a universal vertex. We first prove that maximal weak hubs form a partition of V (D). Proof : Assume X and Y contradicts the claim. If there exists x / ∈ Y which strictly in-dominates X, then there is at least one arc to x from a vertex of X ∩ Y = ∅ and since Y is not mixed, there is no arc from x to Y . As D[Y ] is strongly connected, Lemma 2.4 implies that x strictly in-dominates Y . But then X ∪ Y would be a weak hub, contradicting the maximality of X. If now there exists x ∈ Y which strictly in-dominates X, then consider any arc ab where a ∈ X ∩ Y and b ∈ X \ Y , which must exist since D[X] is strong. But now Y is mixed because of the arcs ab and bx, a contradiction. So X is not strictly in-dominated and is similarly it cannot be strictly out-dominated. Claim 4.5. Every vertex belongs to some maximal weak hub.

Proof : Let u be a vertex of D. It suffices to prove {u} is a weak hub. Assume by contradiction this is not the case. By definition of weak hub this forces u + = u -. If u + = V \ {u}, D is semicomplete with a universal vertex, a contradiction. If not, as D is connected, there exists vertices v ∈ u + and w / ∈ u + such that vw ∈ A or wv ∈ A. But then, as u -= u + , u and w are both in-neighbours or both out-neighbours of v, implying w ∈ u -= u + , a contradiction.

The next claim describes the structure of the arcs linking maximal weak hubs.

Claim 4.6. Let X and Y be two distinct maximal weak hubs. Then either X strictly in-dominates Y , or X strictly out-dominates Y , or there is no arc between X and Y .

Proof : Assume that there exists x in X and y in Y such that xy ∈ A(D). By Lemma 2.4 applied to H = X and z = y, and because X is not mixed by hypothesis, we have that y strictly in-dominates X. But now by applying the same Lemma to the digraph obtained from D by reversing all arcs (which is still locally out-semicomplete since D is locally semicomplete), we get that Y strictly in-dominates X.

we think that a decomposition theorem for locally out-tournament oriented graphs, in the spirit of Theorem 4.2 would be of great help for the general case Finally, we wonder whether Theorem 4.2 can be applied to prove some open problems about locally semicomplete digraphs or locally tournament oriented graphs. One such problem is the famous Second Neighbourhood Conjecture due to Seymour, which is known for tournaments but not for locally tournament oriented graphs. The first two cases of Corollary 4.2 can be dealt with without too much difficulty, but we were alas not able to deal with the last one.

Lemma 2 . 4 .

 24 Let D be a locally out-semicomplete digraph. Let H be a strong subdigraph of D and let z / ∈ H. If z -∩ H = ∅ and z + ∩ H = ∅, then H ⊆ z -.

  D[E], D[F ], D[G] and D[H] are semicomplete; -E strictly out-dominates F , F strictly out-dominates G, G out-dominates H and H outdominates E.

Figure 2 :

 2 Figure 2: The structure in the third case of Theorem 4.2

Claim 4 . 4 .

 44 Let X and Y two distinct maximal weak hubs. Then X ∩ Y = ∅.

Acknowledgment This project was financed by the ANR projects DISTANCIA (ANR-17-CE40-0015), HOSIGRA (ANR-17-CE40-0022) and ALGORIDAM (ANR-19-CE48-0016). We would also like to thanks Reza Naserasr for fruitful discussions.

Let us now consider a partition of V into maximal weak hubs X1, X2 . . . X k and let D the digraph obtained by contracting every set in the partition. By Claim 4.6, D is a locally tournament oriented graph. Moreover, the in-neighbourhood (resp. the out-neighbourhood) of any vertex x ∈ V (D ) is acylic. Indeed, if there were any such cycle C on vertices of D corresponding to weak hubs Xi 1 , Xi 2 . . . Xi with ≥ 2, then ∪ j=1 Xi j would induce a weak hub in-dominated (resp. out-dominated) by every vertex of the weak hub corresponding to x in D, a contradiction to the maximality of the Xi. Due to Theorem 1.1, D is a round digraph.

To prove Theorem 4.2 it remains to prove the following lemma which deals with the case where there exists a maximal weak hub that is mixed. Lemma 4.7. Let D a connected locally semicomplete digraph such that there exists a maximal weak hub which is mixed. Then there exists a partition of V (D) into four sets E, F , G and H such that :

• F and H are non empty, and one of E and G is non empty.

• D[E], D[F ], D[G] and D[H] are semicomplete;

• E strictly out-dominates F , F strictly out-dominates G, G out-dominates H and H out-dominates E.

Proof of Lemma 4.7 : Let X be a maximal weak hub which is mixed. Let us define :

Since X is strong, by Lemma 2.4, X strictly out-dominates X OU T and due to the same lemma applied to the digraph obtained by reversing all arcs of D, we have that X IN strictly out-dominates X.

Eventually since X is assumed to be a weak hub that is mixed, we have that X M and X are non-empty and one of X IN or X OU T is non-empty. Now we claim that for each vertex u in X OU T and each vertex v in X M , uv ∈ A(D). Suppose this is not the case and consider w an in-neighbour of v in X: as u is an out-neighbour of w, there must be an arc between u and v. If vu ∈ A(D) then X ∪ {v} is a weak hub, contradicting the maximality of X. Similarly one can prove that for each vertex v in X IN and for each vertex u in X M , uv ∈ A(D).

There cannot be any arc uv with u ∈ X M ∪ X IN and v ∈ X N O as any such u has an out-neighbour in X and this would create an induce S + 2 . For a similar reason, there cannot be an arc uv with u ∈ X N O and v ∈ X M ∪ X OU T . In particular we prove that there are no arcs between X M and X N O . But now since X M is not empty, consider a vertex w ∈ X M and remember that it out-dominates every vertex in X IN and in-dominates every vertex in X OU T . Now, since D is locally semicomplete, it does not contain S - 2 nor S + 2 , so there cannot be an arc from X OU T to X N O or from X N O to X IN . Therefore there is no arc between

and H = X M , we can verify that at least three of them are non-empty (F , H and one of E and G) and thus that they all are semicomplete (as the in-neighbourhood and the out-neighbourhood of any vertex are semicomplete) and satisfy the output of the theorem.

Two short applications of Theorem 4.2

Let us mention here some applications of the previous theorem.

A 2-king in a digraph is a vertex that can reach every other vertex by a directed path of length at most 2. In [START_REF] Wang | Kings in locally semicomplete digraphs[END_REF] it is proved that a locally semicomplete digraph that is not a blow-up of a round oriented graph (that is, not in the second case of Theorem 4.2) has a 2-king. As we show now, this is a direct consequence of Theorem 3.2. Indeed, when D is semicomplete, it is easy to see that a vertex x of maximum out-degree satisfies that every vertex in x -is either in x + or has an in-neighbour in x + , and therefore x is a 2-king. And if D is described by the third case, we distinguish two cases.

• If H is non empty, then we claim that a vertex x in G that is a 2-king in D[G] (which exists by the previous argument) is a 2-king in D. Indeed, every vertex in H is in x + , and since there is all arcs from H to E, there is a directed path of length at most 2 from x to each vertex of E. Finally, since x has at least one out-neighbor in E and there is all arcs from E to F , there is a directed path of length 2 from x to each vertex in F .

• If H is empty, then F is not and in that case we take any vertex that is a 2-king in E, and it is clearly a 2-king of D.

Another topic is pancyclicity, that is the property that a digraph contains a directed cycle for all possible lengths between 3 and the number of vertices. We don't write the proof here but Theorem 4.2 implies the result of [START_REF] Bang-Jensen | A classification of locally semicomplete digraphs Discrete[END_REF] characterizing pancyclicity for locally semicomplete digaphs, since again the only non easy case is when the digraph is neither semicomplete nor the blowup of a round digraph, in which case it follows without too much effort because of the simplicity of this third case compared to the one of Theorem 3.2 in [START_REF] Bang-Jensen | A classification of locally semicomplete digraphs Discrete[END_REF].

Perspectives

In the context of Conjecture 3.1, it would be interesting to prove that { ← → K 2 , H, S + 2 } is heroic for every hero H. We prove it in the case where the hero consists of a C 3 plus a vertex dominating it. In order to extend this partial result, one idea could be to use the structure theorem for heroes of Berger et

([8])

A tournament is a hero if and only if it can be constructed by the following inductive rules:

• K 1 is a hero.

• If H 1 and H 2 are heroes, then H 1 ⇒ H 2 is also a hero.

• If H is a hero, then for every k ≥ 1, the tournaments C 3 (H, T T k , K 1 ) and C 3 (T T k , H, K 1 ) are both heroes.

In the light of this theorem, a first step to prove that { ← → K 2 , H, S + 2 } is heroic for every hero H would be to show that, given two heroes H 1 and H 2 , if { ← → K 2 , H i , S + 2 } is heroic for i = 1, 2, then { ← → K 2 , H 1 ⇒ H 2 , S + 2 } is also heroic. The case H 1 = K 1 is solved in the prepublication [START_REF] Steiner | On coloring digraphs with forbidden induced subgraphs[END_REF] mentioned in the introduction, but