D Freitas
email: damien.freitas@uca.fr

G Manthilake

Electrical conductivity of hydrous silicate melts: Implications for the bottom-up hydration of Earth's upper mantle

Keywords: electrical conductivity, water storage capacity, hydrous melt, upper mantle, mantle upwelling, mantle transition zone

The upwelling of the hydrous mantle transition zone triggers dehydration-induced partial melting atop the 410-km discontinuity. Here we investigate the electrical conductivity of hydrous silicate melts in the 200-400 km depth range and explore whether melting at the 410-km depths is responsible for the hydration of the upper mantle. Our experimental electrical conductivity data demonstrate that the mantle at 180-350 km depths is mostly melt free, confirming the H2O under-saturated conditions. However, the residual mantle from partial melting atop the 410km discontinuity may contain various possible amounts of water according to the initial mantle transition zone and melt concentrations. This residual H2O could contribute to the hydration of the upper mantle either through diffusion or material replacement by upwelling. Our calculations suggest that the diffusion may not be responsible for the hydration of the upper mantle to present H2O concentration of 50-200 ppm wt.

Melting of the upwelling mantle transition zone with less than 1500 ppm wt. H2O produces residual peridotites with ~ 200 ppm H2O at the 410-km discontinuity. Continuous upwelling of such hydrous residues would gradually replace the dry upper mantle with depleted residual hydrous peridotites in less than 260 Ma. In this study, we propose a bottom-up hydration mechanism for the Earth's upper mantle driven by dehydration-melting at the 410-km discontinuity. The hydrous partial melting at the top of the asthenosphere appears to be a consequence of H2O saturation in the upwelling residual peridotites.

Introduction:

The average H2O content of the Earth's upper mantle is determined to be about 50-200 ppm wt. [START_REF] Hirschmann | Water, melting, and the deep Earth H2O cycle[END_REF][START_REF] Michael | The concentration, behavior and storage of H20 in the suboceanic upper mantle: Implications for mantle metasomatism[END_REF][START_REF] Saal | Vapour undersaturation in primitive mid-ocean-ridge basalt and the volatile content of Earth's upper mantle[END_REF][START_REF] Simons | Volatiles in basaltic glasses from the Easter-Salas y Gomez Seamount Chain and Easter Microplate: Implications for geochemical cycling of volatile elements[END_REF]. These estimates are significantly lower than the H2O storage capacity of peridotite mineral assemblages (Ardia et al., 2012;Demouchy and Bolfan-Casanova, 2016;Férot and Bolfan-Casanova, 2012), indicating water undersaturated conditions for the Earth's upper mantle.

The origin of H2O in the upper mantle is still a subject of debate. However, two possible scenarios have been proposed. In the first hypothesis, the H2O in the upper mantle has been attributed to hydrous residues left behind following partial melting in the upper mantle. In the second scenario, H2O in the upper mantle can be derived from the mixing of dry upper mantle with a hydrous primitive source [START_REF] Hirschmann | Water, melting, and the deep Earth H2O cycle[END_REF].

At present, three distinct partial melting regimes, that could explain the origin of H2O in the upper mantle have been proposed; dehydration melting at the 410-km discontinuity [START_REF] Bercovici | Whole-mantle convection and the transition-zone water filter[END_REF], melting in mantle wedges [START_REF] Hirth | Experimental constraints on the dynamics of the partially molten upper mantle: Deformation in the diffusion creep regime[END_REF] and partial melting of mantle plumes [START_REF] Morgan | Two-stage melting and the geochemical evolution of the mantle : a recipe for mantle plum-pudding[END_REF], all arguing that the residues left behind by the melting events could be responsible for H2O contents in the upper mantle.

While the depleted signature of mid oceanic ridge basalt (MORB)-source upper mantle strongly favors the partial melting hypotheses, the exact nature of the melting regime that led to the hydration of the upper mantle, consistent with geochemical, geophysical and geodynamic observations, remains poorly constrained [START_REF] Hirschmann | Water, melting, and the deep Earth H2O cycle[END_REF].

A recent study based on in situ sound velocity measurements confirmed the dehydration induced melting in the upwelling mantle across the 410-km discontinuity (Freitas et al., 2017). Geophysical observations of high electrical conductivity (up to 1 S/m) [START_REF] Toffelmier | Electromagnetic detection of a 410-km-deep melt layer in the southwestern United States[END_REF] and reduced seismic wave velocities (δVs 4-9%) [START_REF] Revenaugh | Seismic evidance for silicate melt atop the 410-km mantle discontinuity[END_REF][START_REF] Tauzin | Seismic evidence for a global low-velocity layer within the Earth's upper mantle[END_REF] suggest a 50-60 km thick melt layer above the Mantle Transition Zone (MTZ) [START_REF] Revenaugh | Seismic evidance for silicate melt atop the 410-km mantle discontinuity[END_REF][START_REF] Song | Low-velocity zone atop the 410-km seismic discontinuity in the northwestern United States[END_REF][START_REF] Tauzin | Multiple transition zone seismic discontinuities and low velocity layers below western United States[END_REF][START_REF] Tauzin | Seismic evidence for a global low-velocity layer within the Earth's upper mantle[END_REF][START_REF] Vinnik | Low S velocity atop the 410-km discontinuity and mantle plumes[END_REF]. The upwelling of hydrous MTZ with ~ 0.1-0.2 wt.% of H2O (Freitas et al., 2017;[START_REF] Huang | Water content in the transition zone from electrical conductivity of wadsleyite and ringwoodite[END_REF][START_REF] Karato | Water distribution across the mantle transition zone and its implications for global material circulation[END_REF] and the significant differences in H2O storage capacities between the MTZ [START_REF] Bolfan-Casanova | Water partitioning between nominally anhydrous minerals in the MgO-SiO2-H2O system up to 24 GPa: Implications for the distribution of water in the Earth's mantle[END_REF][START_REF] Hirschmann | Storage capacity of H2O in nominally anhydrous minerals in the upper mantle[END_REF][START_REF] Kohlstedt | Solubility of water in the α, β and γ phases of (Mg,Fe)2SiO4[END_REF] and the overlying mantle (Demouchy and Bolfan-Casanova, 2016;Férot and Bolfan-Casanova, 2012) is the governing factor that triggers melting in the upwelling mantle across the MTZ [START_REF] Bercovici | Whole-mantle convection and the transition-zone water filter[END_REF]Freitas et al., 2017). The reported 3.6-9.4 % shear wave velocity drop is compatible with a 0.5-2.5 % melt fraction (Freitas et al., 2017) in the partial melt layer.

The melt produced at the 410-km discontinuity is found to have high H2O contents (Freitas et al., 2017;[START_REF] Matsukage | Density of hydrous silicate melt at the conditions of Earth's deep upper mantle[END_REF]. The mineral-melt equilibrium following melting at the top of the MTZ causes H2O to diffuse from the hydrous melt into solid mineral phases both above and below the melt layer. The H2O diffusion may occur between melt and wadsleyite, transferring H2O back into the underlying MTZ and between meltolivine/pyroxene/garnet, hydrating the overlying mantle. In addition to the diffusion process, the mantle upwells passively away from subduction zones in response to intrusion of slab material into Earth's interior [START_REF] Bercovici | Whole-mantle convection and the transition-zone water filter[END_REF][START_REF] Karato | The Transition-Zone Water Filter Model for Global Material Circulation : Where Do We Stand ? Earth's Deep Water Cycle[END_REF]. The mantle upwelling is significantly faster than H2O diffusion into mantle minerals [START_REF] Bercovici | Whole-mantle convection and the transition-zone water filter[END_REF], replacing the dry upper mantle with residual peridotites. The dehydration melting in the upwelling mantle and subsequent H2O transport mechanisms therefore play a crucial role in water transfer between the upper mantle and the mantle transition zone [START_REF] Bercovici | Whole-mantle convection and the transition-zone water filter[END_REF].

Laboratory measurements of electrical conductivity have been extremely useful in characterizing the material circulation in the Earth's mantle (Dai andKarato, 2009, Yoshino, 2010). The laboratory measurements suggest electrical conductivity is a powerful tool for detecting melt or fluid and their occurrence within the solid mineral matrix at deep mantle conditions [START_REF] Freitas | Simultaneous measurements of electrical conductivity and seismic velocity of partially molten geological materials: Implications for melt fraction in the upper mantle[END_REF][START_REF] Grotenhuis | Melt distribution in olivine rocks based on electrical conductivity measurements[END_REF][START_REF] Manthilake | Dehydration of chlorite explains anomalously high electrical conductivity in the mantle wedges[END_REF][START_REF] Manthilake | Electrical conductivity of lawsonite and dehydrating fluids at high pressures and temperatures[END_REF][START_REF] Maumus | Electrical conductivity and partial melting of mafic rocks under pressure[END_REF]. It is also sensitive to the chemical composition of melt, particularly to volatiles [START_REF] Gaillard | Carbonatite melts and electrical conductivity in the asthenosphere[END_REF][START_REF] Karato | The role of hydrogen in the electrical conductivity of the upper mantle[END_REF][START_REF] Manthilake | Electrical conductivity of wadsleyite as a function of temperature and water content[END_REF][START_REF] Yoshino | Laboratory Electrical Conductivity Measurement of Mantle Minerals[END_REF] and alkalis [START_REF] Ni | Electrical conductivity of dry and hydrous NaAlSi3O8 glasses and liquids at high pressures[END_REF]. In this study, we measure the electrical conductivity of hydrous peridotite at various pressures up to 12 GPa above their melting temperatures. The results were compared with magnetotelluric profiles to gain insight into the extent of melting and melt percolation if hydrous conditions are assumed for the Earth's upper mantle. Based on our results, we discuss different scenarios that could be responsible for the H2O circulation in the upper mantle.

Materials and Methods

Starting materials

Starting materials with a composition similar to KLB-1 peridotite [START_REF] Wang | Subsolidus and melting experiments of K-doped peridotite KLB-1 to 27 GPa: Its geophysical and geochemical implications[END_REF] were prepared using reagent grade oxides, initially mixed in the absence of the required amount of Al(OH)3 to yield the correct mineral composition (ranging from 1.63 to 4.86 mg to obtain 500 mg of peridotite powder with 400 to 1500 ppm H2O, respectively). These powders were dried at 300 °C overnight to remove any adsorbed moisture. Na and K were added as NaCO3 and K2CO3 and then de-carbonated. The decarbonation of the oxide mixtures was carried out by slowly increasing the temperature to 1000°C (1.6°C/min) and keeping it at 1000°C for about 10 hours to ensure complete decarbonation. The decarbonated powder mixture was then cooled to 200°C and stored in a high vacuum furnace at 120°C prior to the hot pressing runs. The required amount of Al(OH)3 was then mixed with the decarbonated powder mixture to obtain the desired amount of H2O in peridotite samples.

Samples with ~0, 500, 700 and 1200 ppm wt. H2O as initial content in powders were prepared in order to have hydrous under saturated conditions at the different targeted pressures. As water solubility increases with pressure in nominally hydrous phases (Ardia et al., 2012;Férot and Bolfan-Casanova, 2012), we always used initial water content under solubility limit at each pressure. However, due to the absorbed moisture in the high pressure synthesis assembly, the water content may be subjected to some possible water (H) diffusion into the samples even with sealed capsules. The X-Rays powder diffraction analyses using a Philips PW 1830 (Cobalt wave-length) and micro-Raman analyses were carried out prior to the electrical conductivity measurements to ensure the absence of additional hydrous phases such as super hydrous Phase-B and partial melts in pre-sintered samples.

Experimental

The resulting powder mixtures were hot pressed to obtain solid sintered samples for electrical conductivity measurements. Hydrous and dry samples were synthesized at different pressures and temperature conditions in a first high pressure run with multi-anvil apparatus, these samples were recovered for in situ measurements that were performed in a second multi-anvil run. The details on samples conditions (pressure, temperature and time at temperature) are given in the table S1. Dry samples were synthesized at target or lower pressures to ensure low water solubility of the material (Ferot and Bolfan 2012, [START_REF] Kohlstedt | Solubility of water in the α, β and γ phases of (Mg,Fe)2SiO4[END_REF]. Hydrous samples were synthesized at target pressure (for 6 and 9 GPa) and 14 GPa.

The synthesis of hydrous samples at 14 GPa performing in situ experiments at 12 GPa allows us to simulate the dehydration induced melting at the 410-km discontinuity (Freitas et al. 2017).

Dry samples and hydrous samples were synthetized for ~2 hours in Gold-Palladium (Au-Pd) capsules and temperatures between 1373 and 1573 K (table S1). These under saturated conditions are required during the synthesis process, in order to avoid hydrous silicate melts along grain boundaries, which could otherwise interfere with the electrical conductivity. Cylindrical core samples ~1.0 mm in length and ~1.2 mm in diameter were prepared from these pre-synthesized sample specimens.

The high-pressure and high-temperature experiments were performed using a 1500ton Kawai-type multi-anvil apparatus at the Laboratoire Magmas et Volcans, Clermont-Ferrand, France. Both sample synthesis and the in situ measurements were performed in multi-anvil apparatus using an octahedral pressure medium composed of MgO and Cr2O3 in a 14/8 multi-anvil configuration (octahedron edge length / anvil truncation edge length) for experiments at 12 GPa and 18/11 for experiments at 6 and 9 GPa (Supplementary Figure S1).

The pre-synthesized cylindrical sample was inserted into a MgO sleeve. The sleeve also helps insulate the sample electrically from the furnace. This furnace, composed of a 25 μm thick cylindrical Re foil, has apertures for the electrode and the thermocouple wires. A zirconia sleeve around the furnace was used as a thermal insulator. Thermal gradient was also taken in to account for the design of these experiments (Hernlund et al., 2006), and were estimated to be negligible <5° with our sample dimensions (Supplementary Figure S2). We placed two electrodes, made of Mo discs, at the top and bottom of the cylindrical sample. A tungstenrhenium (W95Re5-W74Re26) thermocouple junction was placed at one end of the sample to monitor the temperature. The opposite end was connected to a single W95Re5 wire. We collected impedance spectra between the two W95Re5 wires. All ceramic parts of the cell assembly, including the pressure medium, were fired at 1373 K prior to their assemblage in order to remove any adsorbed moisture, and kept in vacuum furnaces (10 -2 Torr and 150°C) before assembling the experiment. Oxygen fugacity of the sample was not controlled during in situ measurements, but was expected to remain below the Mo-Mo2 buffer.

EC measurements were performed using the ModuLab MTS Impedance/Gain-phase analyzer in the frequency range of 10 1 -10 6 Hz. Polyphasic samples are characterized by a combination of resistor-capacitor/constant phase element (R-C/CPE) circuits and the resistance can be obtained by fitting the impedance spectra to appropriate equivalent circuits (Supplementary Figure S3). Once the sample resistance has been determined, conductivity can be calculated using the sample diameter and length. The experimental procedure for electrical conductivity measurements is discussed elsewhere [START_REF] Manthilake | Dehydration of chlorite explains anomalously high electrical conductivity in the mantle wedges[END_REF][START_REF] Manthilake | Electrical conductivity of lawsonite and dehydrating fluids at high pressures and temperatures[END_REF].

Importance of a moisture removal procedure is crucial in these experiments and is detailed in Supplementary Figure S3.

Chemical and micro-textural analyses

The chemical composition and micro-textures of pre-sintered samples and experimental run products after electrical conductivity measurements were investigated using a Cameca SX100 electron probe micro analyzer and a Scanning Electron Microscope (SEM) JEOL Jeol JSM-5910 LV, respectively, at the Laboratoire Magmas et Volcans of Clermont-Ferrand. Energy-dispersive X-ray spectroscopy (EDS) chemical mapping was used to determine the mineral proportions and chemical compositions by selecting a large area of each phase. The fine melt micro-textures were observed using a ZEISS supra 55VP field emission gun (FEG) SEM at 2MATech, Aubière, France. Major element compositions of solid phases are given in Supplementary Tables S2 andS3.

The presence of water (OH) in the crystalline phases of our peridotite samples was qualitatively analyzed both before and after electrical conductivity measurements. These analyzes were performed with micro-Raman spectroscopy (Freitas et al., 2017) (Supplementary Figures S4-S6). Raman spectra were collected using an InVia confocal Raman micro spectrometer, equipped with a 532 nm diode laser (output power of ~140 mW), a Peltier-cooled CCD detector, a motorized XY stage and a Leica DM 2500 M optical microscope, at the Laboratoire Magmas et Volcans, Clermont-Ferrand. The Raman spectroscopy technique was preferred over FTIR analysis mainly due to its capacity to analyze both OH and Si bonds in a non-destructive manner and its high spatial resolution allowing the analysis of small objects (~1 μm). However, due to the unavailability of H2O calibration curves for pyroxene and garnet, we were not able to perform a complete estimate of the bulk H2O contents of our samples (qualitative analysis only).

Results

The melting of hydrous peridotite can be detected by a sudden or discontinuous increase in electrical conductivity at high temperature (Fig. 1). The electrical conductivity of partially molten peridotites varies from 0.1-1 S/m. The activation enthalpies of hydrous peridotites before melting indicate the conduction mechanism is characteristic of electron hopping (small polaron conduction) [START_REF] Katsura | Electrical conductivity of the major upper mantle minerals: a review[END_REF].

The high resolution image analyses of samples after electrical conductivity measurements indicate the presence of melt in hydrous samples (Fig. 2). The experimental run products indicate well developed polygonal textures with average grain size of <30 µm.

The melt distribution appears to be homogeneous throughout the samples and occurs as thin films along grains boundaries and as small pockets of a few hundred nanometers in size at triple junctions (Fig. 2 b,c,d). The melt forms an interconnected network showing a very low dihedral angle with complete wetting of the grain boundaries (Freitas et al., 2017) (Fig 3).

Due to the extremely small size of the melt pockets (<1 µm), we were not able to measure the chemical composition of the resulting hydrous silicate melt at 6 and 9 GPa. The melt fraction appears to be low (up to a few %). The chemical analyses of melt at 12 GPa indicate significant enrichment in incompatibles elements (Ca, Al, Na, K, Fe, Ti) (Freitas et al., 2017).

Samples synthesized prior to electrical conductivity measurements were melt-free (confirmed by X-ray powder diffraction/micro-Raman) and their mineralogy was mainly composed of olivine, clinopyroxene and garnet (Fig. 2a). No evidence of hydrous phases such as super-hydrous B were found in the starting materials. The proportions of the different crystalline phases vary between the samples depending on the degree of melting. The starting samples recovered after synthesis at 12 GPa have mineral proportions of about 60±2% olivine, 20±2% garnet and 20±2% clinopyroxene. Upon partial melting at 12 GPa, the proportion of olivine remained almost constant, while the proportion of Cpx decreased with increasing melt fractions. The proportion of garnet slightly increased with melting, as would be expected with mantle mineralogy [START_REF] Kaminsky | Earth-Science Reviews Mineralogy of the lower mantle : A review of ' super-deep ' mineral inclusions in diamond[END_REF]. For all experiments, the Mg number of olivine is between 90 and 93. Nominally anhydrous sample have lower Mg numbers of 88-89, which is consistent with melt-absent conditions. Chemistry of solid phases for synthesis and experimental runs are given in Supplementary Tables S2 andS3.

Discussion

Melting in the upper mantle

The magnetotelluric and geomagnetic depth sounding profiles of the upper mantle indicate significant regional variation in electrical conductivity [START_REF] Grayver | Joint inversion of satellite-detected tidal and magnetospheric signals constrains electrical conductivity and water content of the upper mantle and transition zone[END_REF][START_REF] Kuvshinov | 3-D modelling and analysis of Dst C-responses in the North Pacific Ocean region, revisited[END_REF][START_REF] Toffelmier | Electromagnetic detection of a 410-km-deep melt layer in the southwestern United States[END_REF]. The majority of geophysical profiles are compatible with electrical conductivity resulting from hydrous upper mantle minerals (Novella et al., 2017b). The comparison of these electrical conductivities of hydrous silicate melts with global electromagnetic and magnetotelluric profiles of the upper mantle implies that the upper mantle in the depth range 180-350 km is mainly melt free (Fig. 4).

However, the estimated high electrical conductivity anomaly above the 410-km discontinuity [START_REF] Toffelmier | Electromagnetic detection of a 410-km-deep melt layer in the southwestern United States[END_REF] appears to be compatible with the presence of melt in this zone (Fig. 4). Thanks to our measurements, the melt fraction required to explain this observed anomaly is estimated to be <10 vol. % (Fig. 4). This estimation is significantly higher than the melt fraction of 0.5-2.5 % obtained by sound velocity measurements (Freitas et al., 2017) to explain the shear velocity reduction (δVs) of 3.6-9.4 %. The possible overestimation of melt fraction by laboratory electrical conductivity methods has been discussed in a recent study [START_REF] Freitas | Simultaneous measurements of electrical conductivity and seismic velocity of partially molten geological materials: Implications for melt fraction in the upper mantle[END_REF] in addition to assumptions made on forward and inversion models of electromagnetic data [START_REF] Toffelmier | Electromagnetic detection of a 410-km-deep melt layer in the southwestern United States[END_REF]. In our discussion, we use our electrical conductivity data to qualitatively determine the vertical extents of melting in the upper mantle.

The absence of melt in the depth range 180-350 km confirms that the water content of the upper mantle is below the saturation limits of peridotite (Ardia et al., 2012;Férot and Bolfan-Casanova, 2012). This water undersaturated mid-upper mantle is sandwiched between two partial melt layers. Melting at the top of the asthenosphere (above 180 km) has been discussed in terms of the reduced water solubility of aluminous orthopyroxene [START_REF] Mierdel | Water Solubility in Aluminous Orthopyroxene and the Origin of Earth's Asthenosphere[END_REF], and melting below 350 km has been linked to the reduced water solubility in peridotites, in both cases melting is caused by the presence of excess H2O. Here we investigate whether the dehydration-induced water saturation and subsequent melting at the top of the mantle transition zone has an influence on water circulation in the Earth's upper mantle.

H2O circulation in the upper mantle

The strong evidences supporting the presence of a melt layer atop the mantle transition zone and the strict requirement of H2O for deep mantle melting underline the importance of understanding the mechanism that is responsible for H2O circulation in the upper mantle. Assuming that the mantle was initially dry and well mixed after the establishment of the vigorously convecting global magma ocean, initiation of subduction in a plate tectonic context can be considered as having been the major process responsible for mantle hydration. With average downward slab velocities of between 5 and 10 cm per year (Gordon, 1995a) and assuming plates thicknesses of 7 km and a total ridge length of 70000 km, the annual amount of subducted material is estimated to be 24-49 km [START_REF] Smyth | Nominally Anhydrous Minerals and Earth's Deep Water Cycle[END_REF]. With a mean age of 80 Ma, the average subducted material is estimated to have between 0.1 and 1.5% of H2O [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF] leading to an H2O flux of 0.1 to 1 km 3 into the mantle per year. However, studies of ophiolites indicate that most hydrous minerals break down at shallow mantle conditions and subducted rocks in eclogite facies were shown to be significantly impoverished in volatiles, with values of up to 0.25% H2O [START_REF] Katayama | Hydroxyl in clinopyroxene from the deep subducted crust : Evidence for H2O transport into the mantle[END_REF]. Even in the case of extreme slab dehydration of up to 97% [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF], H2O contents of up to 1000 ppm wt. can be preserved in subducting slabs and transported down to the transition zone [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF].

Using slab downward speed of 5 cm per year and the MTZ thickness of 250 km, the time required for a portion of slab to cross the MTZ is estimated to be 5 Ma. While this time is relatively short on a planetary timescale, the slab deflection and stagnation in the MTZ observed in seismic tomography studies [START_REF] Zhao | Global tomographic images of mantle plumes and subducting slabs : insight into deep[END_REF] suggests that residence time of subducted material would be much greater than the values assumed in our calculations. With a diffusion coefficient of 10 -8 -10 -10 m²/s [START_REF] Hae | Hydrogen diffusivity in wadsleyite and water distribution in the mantle transition zone[END_REF][START_REF] Richard | Slab dehydration in the Earth ' s mantle transition zone[END_REF], a significant transfer of H2O from the slab to the MTZ could be expected during that time.

The appearance of the melt layer above the MTZ can be inferred given the estimation of H2O slab fluxes in the mantle transition zone. If we assume 24.5 km 3 slab material reaches the MTZ per year (volume produced at the ridges for 5 cm/year spreading velocity along 70000 km of ridges with an average thickness of 7 km (Gordon, 1995b)), and that these rocks contain 1000 ppm wt. H2O, a mass of 9.4×10 10 kg of H2O is transferred to the MTZ every year. Assuming the volume of the MTZ to be 6.5×10 7 km 3 , with a mass of 2.6×10 20 kg (mean density of 4000 kg/m 3 ), and assuming complete slab dehydration into the MTZ, only 2.2-3.3 Ma after the beginning of modern subduction is required to reach 800-1200 ppm wt. in the MTZ, which is roughly equal to the maximum H2O solubility in nominally anhydrous minerals (NAM's) forming the upper mantle (Ardia et al., 2012;Demouchy and Bolfan-Casanova, 2016;Férot and Bolfan-Casanova, 2012). Using the upper limit of the inferred H2O content of the mantle transition zone from geophysical methods of about 0.1-0.2 wt.% H2O (Freitas et al., 2017;[START_REF] Huang | Water content in the transition zone from electrical conductivity of wadsleyite and ringwoodite[END_REF], the time required to transport H2O from surface to the MTZ is estimated to be 6 ± 0.6 Ma. This very simple estimation highlights the efficiency of the hydration of the MTZ. It indicates clearly that melting at the 410-km discontinuity would have been initiated early in the Earth's history. However, H2O diffusion into wadsleyite is a slow process [START_REF] Hae | Hydrogen diffusivity in wadsleyite and water distribution in the mantle transition zone[END_REF]. For example, the time required to diffuse 2000 ppm wt. H2O to reach 500 ppm wt. at 500 km laterally along the MTZ is estimated to be 175 Ga. Given that the surface area of the MTZ is significantly large and the number of active subduction zones is extremely limited, it is possible that the H2O distribution in the MTZ remains heterogeneous, with regions of the MTZ maintaining dry conditions [START_REF] Yoshino | Dry mantle transition zone inferred from the conductivity of wadsleyite and ringwoodite[END_REF]. The absence of seismic anomalies in some regions [START_REF] Tauzin | Seismic evidence for a global low-velocity layer within the Earth's upper mantle[END_REF] indicates that melting is not pervasive above the MTZ. The upwelling of H2O deprived regions (< 800-1200 ppm wt.) of the MTZ may not undergo partial melting upon entering the upper mantle.

The H2O under-saturated MTZ, as demonstrated by recent experimental studies (Freitas et al., 2017;[START_REF] Huang | Water content in the transition zone from electrical conductivity of wadsleyite and ringwoodite[END_REF][START_REF] Karato | Water distribution across the mantle transition zone and its implications for global material circulation[END_REF], can be a result of continuous removal of H2O from the MTZ (upward and/or downward) or partial dehydration of slabs at the MTZ.

The first scenario implies the presence of a global H2O circulation cycle extracting H2O from the MTZ from an early stage. However, the quantity of H2O stored in the melt layer above the MTZ together with the moderately hydrous olivine present in equilibrium with melt over the 60 km thick layer, only represents a small portion of the total amount of H2O in the MTZ (<1%). This amount is a few orders of magnitude lower than the total amount transported by subducted slabs per year. Similarly, the known deep volcanic sources of OIB show that source material may be only moderately hydrous with H2O content estimated between 300 and 1000 ppm wt. [START_REF] Rüpke | Implications of Subduction Rehydration for Earth ' s Deep Water Cycle[END_REF][START_REF] Simons | Volatiles in basaltic glasses from the Easter-Salas y Gomez Seamount Chain and Easter Microplate: Implications for geochemical cycling of volatile elements[END_REF]. These calculations demonstrate the possibility of large scale melting events occurring periodically in the Earth's history.

Large scale deep mantle melting has occurred in the past, as demonstrated by the presence of komatiites, kimberlites and numerous large igneous provinces [START_REF] Vinnik | Low S velocity atop the 410-km discontinuity and mantle plumes[END_REF]. However, at present, such large scale volcanism does not exist anymore and hot spot volcanism and mantle plumes are alternative candidates for removing H2O from the deep Earth.

On the other hand, it also appears possible that slabs that are not fully dehydrated at the MTZ, and instead carry a substantial amount of H2O on down to the lower mantle [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF]. The high H2O contents of up to 2 wt. % observed in some enriched mantle sources of oceanic island basalts (OIB) indicate that some H2O may be incorporated into dense hydrous silicate phases [START_REF] Nishi | Stability of hydrous silicate at high pressures and water transport to the deep lower mantle 7[END_REF][START_REF] Pamato | Lower-mantle water reservoir implied by the extreme stability of a hydrous aluminosilicate[END_REF] that penetrates down into the lower mantle via subduction processes [START_REF] Rüpke | Implications of Subduction Rehydration for Earth ' s Deep Water Cycle[END_REF]. These would imply a slower hydration rate of the MTZ and more tardive apparition of the LVL.

Bottom-up hydration of the upper mantle

In this section, different possible mechanisms of bottom-up hydration of Earth's upper mantle and their consequences are explored.

Due to the H2O under-saturated conditions of the upper mantle, the water contained in hydrous melts produced at the 410-km discontinuity may diffuse in to the overlying mantle.

Based on the one dimensional diffusion equation [START_REF] Crank | The mathematics of diffusion[END_REF], we calculate the diffusion time required to hydrate the upper mantle assuming that the upper mantle is initially dry (Supplementary Figure S7),

𝑪(𝒙, 𝒕) = 𝑪 𝟎 + (𝑪 𝟏 -𝑪 𝟎 ) * 𝒆𝒓𝒇𝒄 ( 𝒙 𝟐√𝑫 * 𝒕 )

where C0 is the initial concentration, C1 is constant local concentration, x is the distance to the border of concentration C1, t is the diffusion time and D the diffusion coefficient of hydrogen in olivine. The value for D was estimated to be 7.82×10 -9 ±6.1×10 -8 m²/s at 1723 K for a randomly oriented polycrystalline material (average of the three crystallographic direction) (Novella et al., 2017b). Using average H2O solubility for mantle (Demouchy and Bolfan-Casanova, 2016;Férot and Bolfan-Casanova, 2012), increasing from 300 ppm wt. at 200 km up to 1000 ppm wt. at 410 km, we calculate the time required for peridotite in equilibrium with the hydrous melt layer to reach H2O saturation. Our results indicate that the time required to reach the H2O saturation in the uppermost mantle is unrealistically high based on diffusion alone, and cannot be achieved during the planetary life time (Supplementary Figure S7, Fig. 5a).

On the other hand, the mantle away from subducting slabs is rising passively in response to the material injection by subduction. Estimated values using slab flux yield to upwelling velocities of 1 mm per year [START_REF] Bercovici | Whole-mantle convection and the transition-zone water filter[END_REF]. The estimation based on convection models indicates convection rate as high as 1.5 cm/year [START_REF] Becker | Thermal constraints on the survival of primitive blobs in the lower mantle[END_REF]). The upwelling rates from subduction fluxes indicates that only 260 Ma is sufficient to bring rocks from the top of the transition zone to the uppermost mantle (150 km depth) and even faster considering the highest convection rates. This estimation indicates that residual hydrous peridotite should be rising from the LVL to the uppermost mantle and saturates the upper mantle at various depths depending on the initial H2O contents in peridotite (Fig. 5b).

Here we investigate the possible H2O contents of peridotites in contact with the melt layer. The shear wave velocity reduction (δVs) at the melt layer is estimated to vary between 3.6-9.4 % [START_REF] Revenaugh | Seismic evidance for silicate melt atop the 410-km mantle discontinuity[END_REF][START_REF] Tauzin | Seismic evidence for a global low-velocity layer within the Earth's upper mantle[END_REF]. Based on our previous study (Freitas et al., 2017), the resulting melt fraction corresponding to the δVs at the low velocity layer is estimated to be 0.5-2.5 vol.%

The average water content of the MTZ has been constrained by different methods with variable results. Despite potential high water solubilities of MTZ minerals (Bolfan-Casaonova et al., 2018b), the actual content of the MTZ is believed to be lower than saturation limit. Water contents inferred from inclusions in diamonds [START_REF] Pearson | Hydrous mantle transition zone indicated by ringwoodite included within diamond[END_REF] with up to 2 % would suggest potentially high water content which also supported by viscosity measurements [START_REF] Fei | A nearly water-saturated mantle transition zone inferred from mineral viscosity[END_REF]. Particularly, the near-water saturated conditions implied by hydrous ringwoodite inclusions found in natural diamond [START_REF] Pearson | Hydrous mantle transition zone indicated by ringwoodite included within diamond[END_REF] may represent an episode of subduction-zone-related melting. However, most of geophysical techniques such as elasticity, sound velocities [START_REF] Inoue | Elastic properties of hydrous ringwoodite (?-phase) in Mg2SiO4[END_REF][START_REF] Chang | Comparative compressibility of hydrous wadsleyite and ringwoodite: effect of H2O and implications for detecting water in the transition zone[END_REF], Freitas et al., 2017) and electrical conductivities [START_REF] Huang | Water content in the transition zone from electrical conductivity of wadsleyite and ringwoodite[END_REF][START_REF] Yoshino | Dry mantle transition zone inferred from the conductivity of wadsleyite and ringwoodite[END_REF][START_REF] Yoshino | Re-evaluation of electrical conductivity of anhydrous and hydrous wadsleyite[END_REF] indicating major part of MTZ contains < 0.2 wt. % H2O.

Assuming a batch melting scenario and H2O partitioning between peridotite and melt (Novella et al., 2014), readjusted to the mineral proportions expected at 12 GPa (Supplementary Text S1), we could estimate the H2O contents of peridotite in equilibrium with hydrous melt for variable water contents in the mantle source for different melt fractions (Fig. 6). Different scenarios could occur given the potential range of water content of the residual peridotites (Fig. 6). Thus, if residual peridotite in the upwelling mantle contains about 200 ppm wt. of H2O, mantle saturation can be expected at range of depths in the upper mantle (Fig. 6). If we assume the H2O saturation curves of Férot and Bolfan-Casanova (Férot and Bolfan-Casanova, 2012), Hirschmann et al. [START_REF] Hirschmann | Dehydration melting of nominally anhydrous mantle: The primacy of partitioning[END_REF], Tenner et al. [START_REF] Tenner | H2O storage capacity of olivine and low-Ca pyroxene from 10 to 13 GPa: Consequences for dehydration melting above the transition zone[END_REF][START_REF] Tenner | Hydrogen partitioning between nominally anhydrous upper mantle minerals and melt between 3 and 5 GPa and applications to hydrous peridotite partial melting[END_REF] and Ardia et al. (Ardia et al., 2012) (for garnet/olivine partitioning of H2O of 9.0), peridotite in the upper mantle reaches H2O saturation limits above 200 km. In contrast, by using the extreme H2O saturation scenario proposed by Ardia et al. (Ardia et al., 2012) with water partitioning between olivine and garnet of 0.9, the mantle saturation above 200 km depths can be expected for the water contents of 450 ppm. However, this value is inconstant with the upper mantle water contents of 50-200 ppm.

The ascent of residual hydrous peridotites would initiate melting upon reaching their storage capacities and could be responsible for the hydrous melting at the top of the asthenosphere. Melting above a depth of 180 km can only be explained by moderate water content in the MTZ (< 1500 ppm wt) for geologically relevant melt fractions at the 410-km discontinuity (Fig. 6). The upwelling of the MTZ with water contents of less than 800 ppm wt. may not undergo melting upon entering the upper mantle and may continue to hydrate the upper mantle. However, such fertile peridotites are not able to explain the depleted signature of the upper mantle. Melting at the 410-km discontinuity is therefore a primary requirement for a compatible hydration of the upper mantle with geochemical constrains.

The lithosphere-asthenosphere boundary (LAB) is defined as intersection of the conductive geotherm of mechanically strong crust with the fluid-saturated peridotite solidus in mechanically weak upper most part of the mantle [START_REF] Mckenzie | The Volume and Composition of Melt Generated by Extension of the Lithosphere[END_REF]. The location of the LAB is therefore corresponding to the depth at which the fluid saturation occurs in upwelling peridotites. The geochemical constraints suggest that the LAB is not a stable feature, which appear to move up or down over geological time scale [START_REF] O'reilly | The continental lithosphere-asthenosphere boundary: Can we sample it?[END_REF]. If other parameters remain constant, the lateral variation of the depth of the LAB may indicate the variable water contents in residual peridotites, which corresponds to the lateral variations of water contents in the MTZ. Further, the movement of the depth of the LAB at the same location may suggest episodic variation of water contents in the MTZ over geological time scale. The upwelling of MTZ with a water content greater than 2000 ppm wt.

ppm could produce residual peridotites reaching the maximum water storage capacity (Fig. 6). The upwelling of such peridotites could continue to melt and form a gravitationally unstable melt column in the upper mantle. The upwelling of the MTZ with high water contents would trigger large-scale melting [START_REF] Vinnik | Low S velocity atop the 410-km discontinuity and mantle plumes[END_REF] and could be responsible for the formation of upper mantle plumes.

Conclusions

Based on our data, we propose a hydration mechanism for the Earth's upper mantle that is consistent with geophysical observations (magnetotelluric and seismic profiles) and geochemical constraints. In this bottom-up water transportation mechanism, the residual hydrous peridotites, formed as a result of melting at the 410-km discontinuity, upwell across the upper mantle, replacing initially dry mantle peridotites. The resulting water content in the residues is directly related to the water content of the mantle transition zone and the degree of melting. The upper limit of 200 ppm observed for the upper mantle can be maintained in residual peridotites if the water content of the MTZ is less than 1500 ppm . The slow H2O diffusion rates in wadsleyite may enforce large gradient in H2O distribution within the MTZ, and as a result, regional variations of H2O contents in the upper mantle would be expected.

The ascent of residual hydrous peridotites could be responsible for the hydrous melting at the top of the asthenosphere. The uncertainties in the estimation of the electrical conductivity result from the estimations of temperature, pressure, and sample dimensions as well as data fitting errors and are less than 5%. Dihedral angles were estimated using FEG SEM and SEM image analysis with Matlab code to measure the angles by the tangent method. Analytical error is estimated to be 5° (1σ).

Relevant literature values for similar or analogue systems at lower pressures have been represented for comparison [START_REF] Mibe | Control of the location of the volcanic front in island arcs by aqueous fluid connectivity in the mantle wedge[END_REF][START_REF] Mibe | Connectivity of aqueous fluid in the Earth's upper mantle[END_REF][START_REF] Watson | Continental Mantle[END_REF][START_REF] Yoshino | Complete wetting of olivine grain boundaries by a hydrous melt near the mantle transition zone[END_REF]. samples with dashed lines. The conductivity profiles are for oceanic mantle (pacific, KO5) [START_REF] Kuvshinov | 3-D modelling and analysis of Dst C-responses in the North Pacific Ocean region, revisited[END_REF], continental (Tuscan SW USA, T07) [START_REF] Toffelmier | Electromagnetic detection of a 410-km-deep melt layer in the southwestern United States[END_REF] and global electrical conductivity model of Earth's mantle based on inverting satellite magnetic field measurements (G17) [START_REF] Grayver | Joint inversion of satellite-detected tidal and magnetospheric signals constrains electrical conductivity and water content of the upper mantle and transition zone[END_REF]. Pink shaded area indicates the possible distribution of hydrous melt at the top of the mantle transition zone. In this area the high conductivity anomaly modeled from EM data has been represented [START_REF] Toffelmier | Electromagnetic detection of a 410-km-deep melt layer in the southwestern United States[END_REF]. Dry upper mantle electrical conductivity vs depth profile based on high pressure electrical conductivity measurements are also displayed [START_REF] Yoshino | Dry mantle transition zone inferred from the conductivity of wadsleyite and ringwoodite[END_REF].

Electrical conductivities of oriented single crystal olivine from measurements [START_REF] Dai | High and highly anisotropic electrical conductivity of the asthenosphere due to hydrogen diffusion in olivine[END_REF]) (4 GPa and extrapolated to mantle temperatures) and diffusion experiments (Novella et al., 2017b) (2 GPa, extrapolated to mantle temperatures and with 80 ppm of water) have been reported. Electrical conductivity anomaly of 350' cannot be reproduced by moderately anisotropic solid rock but rather require presence of melts. Our solid hydrous peridotites conductivities agree well with randomly oriented olivine values. concentration of 800 ppm wt. and variable diffusion lengths (from 410-km to the saturation depth at a given water content) (see also Supplementary Fig. S7). The water saturation curves are for peridotite without considering garnet (F12) (Férot and Bolfan-Casanova, 2012) and considering water partition between olivine and garnet (0.9) (A12) (Ardia et al., 2012). The Table S3. Average chemical composition of solid mineral phases at 12 GPa. Each value is an average of 5 to 20 analyses. The standard deviation is given on the average value in parenthesis.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. High resolution microphotographs of the recovered samples before and after

Figure 3 .

 3 Figure 3. Dihedral angles estimated in our partially molten samples as a function of

Figure 4 .

 4 Figure 4. Electrical conductivity of hydrous silicate melts compared with geophysical

Figure 5 .

 5 Figure 5. Comparison of water transport mechanisms of the Earth's upper mantle. (a)

  broken lines indicate the time required to saturate the upper mantle by diffusion at given depths for a given bulk water content. The shaded in yellow indicates regions where dehydration-melting is expected. The shaded in pink indicate depth range for mantle H2O saturation which can't be explained by diffusion process. (b) Hydration by upwelling. The dehydration melting at the 410-km discontinuity(Freitas et al., 2017) produces residual peridotites with different H2O concentration according melt fractions and compositions. The continuous upwelling of hydrous peridotite replaces the initially dry peridotite in the upper mantle, forming hydrous upper mantle. The ascending peridotite reaches water saturation levels at different depths(Ardia et al., 2012; Férot and Bolfan-Casanova, 2012), and should initiate hydrous melting. The broken lines indicate the time required to saturate the upper mantle at given depths for a given bulk water content. The water saturation curves are for peridotite without considering garnet (F12)(Férot and Bolfan-Casanova, 2012) and with considering water partition between olivine and garnet (A12)(Ardia et al., 2012). The shaded in yellow indicates regions where dehydration-melting is expected. The shaded in pink indicate depth range for mantle H2O saturation.

Figure 6 .

 6 Figure 6. Bottom-up hydration of the upper mantle. The H2O contents are calculated

Total 99.88 (0.93) 100.16 (1.14) 97.86 (5.21) 98.94 (0.67) 99.50 (1.40) 98.62 (1.05) 99.38 (0.45) 99.39 (0.57) 99.95 (1.70) 100.70 (0.29) 100.68 (0.61) 100.71 (1.03) Ol:olivine, Cpx: clino-pyroxenes, Gt: garnet

		M412-7 (0.27)% Melt		M421-0% Melt		M425-2 (0.45)% Melt	M456-3.5 (2.44)% Melt
		Ol	Cpx	Gt	Ol	Cpx	Gt	Ol	Cpx	Gt	Ol	Cpx	Gt
	SiO2	41.60 (0.38)	56.48 (1.01)	47.23 (0.89)	40.90 (0.87)	54.62 (0.65)	40.69 (1.38) 41.36 (0.26)	54.78 (0.67)	43.95 (1.46)	41.99 (0.48)	55.73 (0.36)	46.51 (4.01)
	TiO2	0.00 (0.00)	0.03 (0.01)	0.78 (0.38)	0.04 (0.03)	0.10 (0.04)	0.57 (0.39)	0.04 (0.03)	0.06 (0.03)	0.43 (0.16)	0.05 (0.04)	0.10 (0.05)	0.32 (0.26)
	Al2O3	0.12 (0.02)	2.66 (0.29)	8.87 (2.21)	0.04 (0.08)	1.66 (0.46)	22.19 (3.04)	0.44 (0.60)	1.74 (0.57)	17.13 (5.31)	0.15 (0.04)	1.83 (0.40)	13.17 (10.42)
	Cr2O3	0.12 (0.02)	0.01 (0.02)	0.01 (0.01)	0.01 (0.02)	0.01 (0.02)	0.03 (0.02)	0.01 (0.01)	0.02 (0.01)	0.03 (0.02)	0.00 (0.00)	0.00 (0.02)	0.00 (0.00)
	FeO	7.15 (0.24)	4.83 (0.48)	9.77 (1.77)	8.99 (1.12)	2.97 (0.71)	7.73 (0.58)	7.01 (0.78)	2.90 (0.35)	6.34 (0.61)	5.55 (1.96)	2.96 (0.15)	5.17 (0.99)
	MnO	0.02 (0.02)	0.02 (0.02)	0.05 (0.02)	0.02 (0.02)	0.01 (0.01)	0.06 (0.02)	0.02 (0.02)	0.02 (0.02)	0.06 (0.03)	0.01 (0.01)	0.02 (0.01)	0.05 (4.59)
	MgO	50.78 (0.58)	33.61 (0.73)	13.64 (8.72)	48.80 (0.68)	18.74 (5.83)	18.51 (4.17) 50.10 (1.20)	19.09 (1.18)	24.97 (7.53)	52.67 (2.02)	17.83 (0.62)	24.57 (4.59)
	CaO	0.18 (0.03)	1.70 (0.23)	15.99 (4.72)	0.11 (0.12)	20.04 (4.33)	8.77 (7.09)	0.36 (0.30)	19.79 (1.07)	6.94 (1.97)	0.23 (0.14)	20.96 (0.50)	10.58 (2.07)
	Na2O	0.02 (0.02)	0.21 (0.13)	1.46 (0.42)	0.02 (0.04)	1.31 (0.29)	0.06 (0.02)	0.03 (0.02)	0.97 (0.18)	0.11 (0.07)	0.03 (0.02)	1.23 (0.12)	0.31 (0.25)
	NiO	0.01 (0.01)	0.07 (0.02)	0.00 (0.00)	-	-	-	-	-	-	0.01 (0.01)	0.00 (0.00)	0.02 (0.01)
	K2O	0.01 (0.01)	0.01 (0.02)	0.06 (0.06)	0.01 (0.01)	0.03 (0.02)	0.01 (0.02)	0.00 (0.00)	0.01 (0.01)	0.01 (0.01)	0.01 (0.01)	0.02 (0.01)	0.02 (0.01)

Acknowledgements

We thank F. Schiavi for the assistance with Raman analyses, A. Mathieu for technical assistance, J.M Hénot, J.L Devidal and 2MAtech staff for their help in SEM and EMP and FEG-SEM analyses, respectively. We appreciate the discussion with J, Chantel, D, Andrault, N. Bolfan-Casanova and M.A Bouhifd. G.M. acknowledges funding from the French PNP program (INSU-CNRS). This research was financed by the French Government Laboratory of Excellence initiative n°ANR-10-LABX-0006, the Région Auvergne and the European Regional Development Fund. This is ClerVolc contribution number xx.

Supplementary materials for:

Electrical conductivity of hydrous silicate melts: Implications for the bottom-up hydration of Earth's upper mantle D.Freitas 1* , G. Manthilake 1 1 Université Clermont Auvergne, CNRS, IRD, OPGC, Laboratoire Magmas et Volcans, F-63000 Clermont-Ferrand, France (*corresponding author: damien.freitas@uca.fr)

Supplementary materials

Here we give supplementary information concerning:

 The estimation of the H2O partition coefficient for peridotite/melt at 12 GPa (Text S1)  Experimental conditions and parameters of syntheses and in situ measurements (table S1)  Average chemical composition of solid mineral phases at 6 and 9 GPa both before and after EC measurements (Table S2)

 Average chemical composition of solid mineral phases at 12 GPa. (Table S3) The H2O partition coefficient between peridotite and melt is available at 6 GPa (Novella et al., 2014). Due to the significant variation of mineral proportions at 12 GPa, we have estimate the H2O partition coefficient between peridotite and melt based on mantle mineralogy at 12 GPa. The bulk partition coefficient can be calculated by:

Where X is the fraction of each mineral (i) and Kd its partition coefficient for water with melt.

The Kd values are taken from Novella et al. 2014(Novella et al., 2014) Partially molten samples characteristics are given, with melt fractions and estimated melting temperatures based on electrical conductivity. Dihedral angles measurements are given with average and median angles for each partially molten sample. In the last column (remarks), the type of each experiment is given; synthesis run number used for electrical conductivity measurements is given in parenthesis. Errors given in parenthesis are 1 standard deviation.

1 Water content of the bulk sample estimated with quenched glass quantification and partition coefficient (Novella et al. 2014) with mineral proportions inferred from BSE image analyses. 2 Melt fractions estimated by mass balance, partition coefficients and image surface analyses.

3 Melt fractions estimated by image surface analysis only. 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) Cr2O3 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) .17 (0.12) MnO 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) NiO 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 2006 (Hernlund et al., 2006) . The maximum temperature gradient of the assembly at 1500 K was estimated to be less than 20 K. (Demouchy and Bolfan-Casanova, 2016;Férot and Bolfan-Casanova, 2012) and with considering water partition between olivine and garnet (0.9) (A12) (Ardia et al., 2012). The diffusion profiles even for Earth's time scale can't reach the saturation curve.

Diffusion is able to keep large gradients of water concentration in Earth's upper mantle over geological times.


François Dross

Pascal Ochem

Vertex partitions of (C 3 , C 4 , C 6 )-free planar graphs

published or not. The documents may come

Vertex partitions of (C3, C4, C6) -free planar graphs

Introduction

A graph is (k 1 , k 2 )-colorable if its vertex set can be partitioned into a graph with maximum degree at most k 1 and and a graph with maximum degree at most k 2 . Choi, Liu, and Oum [START_REF] Choi | Characterization of cycle obstruction sets for improper coloring planar graphs[END_REF] have established that there exists exactly two minimal sets of forbidden cycle length such that every planar graph is (0, k)colorable for some absolute constant k.

• planar graphs without odd cycles are bipartite, that is, (0, 0)-colorable.

• planar graphs without cycles of length 3, 4, and 6 are (0, 45)-colorable.

The aim of this paper is to improve this last result. Notice that forbidding cycles of length 3, 4, and 6 as subgraphs or as induced subgraphs result in the same graph class. For every n 3, we denote by C n the cycle on n vertices. So we are interested in the class C of (C 3 , C 4 , C 6 )-free planar graph.

We will prove the following two theorems in the next two sections.

Theorem 1. Every graph in C is (0, 6)-colorable.

Theorem 2. For every k 1, either every graph in C is (0, k)-colorable, or deciding whether a graph in C is (0, k)-colorable is NP-complete.

In addition, we construct a graph in C that is not (0, 3)-colorable in Section 4. This graph and Theorem 2 imply the following. Corollary 3. Deciding whether a graph in C is (0, 3)-colorable is NP-complete.

Since we deal with (0, k)-colorings for some k 2, we denote by the letter 0 the color of the vertices that induce the independent set and we denote by the letter k the color of the vertices that induce the graph with maximum degree k.

Proof of Theorem 1

The proof will be using the discharging method. For every plane graph G, we denote by V (G) the set of vertices of G, by E(G) the set of edges of G, and by F (G) the set of faces of G.

Let us define the partial order . Let n 3 (G) be the number of 3 + -vertices in G. For any two graphs G 1 and G 2 , we have G 1 ≺ G 2 if and only if at least one of the following conditions holds:

• |V (G 1 )| < |V (G 2 )| and n 3 (G 1 ) n 3 (G 2 ). • n 3 (G 1 ) < n 3 (G 2 ).
Note that the partial order is well-defined and is a partial linear extension of the subgraph poset.

We suppose for contradiction that G is a graph in C that is not (0, 6)-colorable and is minimal according to . Let n denote the number of vertices, m the number of edges and f the number of faces of G. For every vertex v, the degree of v in G is denoted by d(G). For every face α, the degree of α, denoted d(α), is the number of edges that are shared between this face and another face, plus twice the number of edges that are entirely in α. More generally, when counting the number of edges of a certain type in a face, we will always count twice the edges that are only in this face. For all d, let us call a vertex of G of degree d, at most d, and at least d a d-vertex, a d --vertex, and a d + -vertex respectively. For all vertex v, a d-neighbor, a d --neighbor, and a d + -neighbor of v is a neighbor of v that is a d-vertex, a d --vertex, and a d + -vertex respectively. For all d, let us call a face of G of degree d, at most d, and at least d a d-face, a d --face, and a d + -face respectively. For all set S of vertices, an S-vertex is a vertex that belongs to S, and an S-neighbor of a vertex v is a neighbor of v that belongs to S. For all set S of vertices, let G[S] denote the set of vertices induced by S, and G -S = G[V (G) \ S]. For convenience, we will note G -v for G -{v}.

Let us first prove some results on the structure of G, and then we will prove that G cannot exist, thus proving the theorem.

Lemma 4. G is connected.

Proof. If G is not connected, then every connected component of G is smaller than G and thus admits a (0, 6)-coloring. The union of these (0, 6)-colorings gives a (0, 6)-coloring of G, a contradiction. Lemma 5. G has no 1-vertex.

Proof. Let v be a 1-vertex and w be the neighbor of v. The graph G -v admits a (0, 6)-coloring since G -v ≺ G. We get a (0, 6)-coloring G by assigning to v the color distinct from the color of w, a contradiction. Lemma 6. Every 7 --vertex of G has a 8 + -neighbor.

Proof. Let v be a 7 --vertex with no 8 + -neighbors. The graph G -v admits a (0, 6)-coloring since G -v ≺ G. If there is a neighbor w of v with no neighbor colored 0, then we color w with 0. Thus, we can assume that every neighbor of v that is colored k has a neighbor colored 0 in G -v, and thus at most 5 neighbors colored k in G -v. Also, we can assume that v has at least one neighbor colored 0, since otherwise v can be colored 0. Thus, v has at most 6 neighbors colored k and v can be colored k, a contradiction.

Lemma 7. Every vertex with degree at least 3 and at most 7 has two 8 +neighbors.

Proof. Suppose for contradiction that G contains a d-vertex v such that 3 d 7 and such that v has at most one 8 + -neighbor. By Lemma 6, v has exactly one 8 + -neighbor w. Let w 1 , . . . , w d-1 be the other neighbors of v. Let H be the graph obtained from G -v by adding d -

1 2-vertices v 1 , . . . , v d-1 , such that for every i ∈ {1, d -1}, v i is adjacent to w and w i . Notice that H ≺ G since n 3 (H) = n 3 (G)-1. Moreover, every cycle of length in H is associated a cycle of length or -2 in G. Therefore H ∈ C, so H has a (0, 6)-coloring. If w is colored 0, then every v i is colored k, coloring v with k leads to a (0, 6)-coloring of G, a contradiction. Therefore w is colored k.
While at least one of the w i 's has no neighbor colored 0 in G -v, we color it 0, and color the corresponding v i with k if it was colored 0. By doing this, we keep a (0, 6)-coloring of H. We can thus assume that in G -v, every w i that is colored k has a neighbor colored 0 and thus at most five neighbors colored k. If at least one of the v i 's is colored k, then w has at most five neighbors colored k in G -v, and assigning k to v gives a (0, 6)-coloring of G. Otherwise, every v i is colored 0, every w i is colored k, and w is colored k. Thus we assign 0 to v to obtain a (0, 6)-coloring of G, a contradiction.

Lemma 8. No 3-vertex is adjacent to a 2-vertex.
Proof. Let w be a 3-vertex adjacent to a 2-vertex v, let x 1 and x 2 be the other two neighbors of w, and let u be the other neighbor of v. Let H be the graph obtained from G -{v, w} by adding five 2-vertices v 1 , v 2 , w 1 , w 2 , and x which form the 8-cycle uv 1 w 1 x 1 xx 2 w 2 v 2 . It is easy to check that H is in C. By Lemmas 6 and 7, u, x 1 , and x 2 are 8 + -vertices in G and thus are 9 + -vertices in

H. Since w is in G but not in H, n 3 (H) = n 3 (G) -1, so H ≺ G. Therefore H has a (0, 6)-coloring.
Suppose that v 1 and v 2 are both colored 0. Then w 1 , w 2 , and u are colored k. We color v with 0 and w with k. The number of neighbors of x 1 (resp. x 2 ) colored k in G is at most the number of neighbors of x 1 (resp. x 2 ) colored k in H. Thus we have a (0, 6)-coloring of G, a contradiction. Now we assume without loss of generality that v 1 is colored k. We color w with the color of x and we color v with k. The number of neighbors of u (resp. A special face is a 5-face with three 2-vertices and two non-adjacent 8 +vertices. See figure 1, left. A special configuration is three 5-faces sharing a common 3-vertex adjacent to three 8 + -vertices, such that all the other vertices of these faces are 2-vertices. See figure 1, right. We say special structure to speak indifferently about a special face or a special configuration.

x 1 , x 2 ) colored k in G is at most the number of neighbors of u (resp. x 1 , x 2 ) colored k in H. Thus we have a (0, 6)-coloring of G, a contradiction. u v 0 v 1 x 0 y 0 u v 0 v 1 x 0 y 0 v 2 x 1 y 1 x 2 y 2
Let us define a hypergraph G whose vertices are the 8 + -vertices of G and the hyperedges correspond to the sets of 8 + -vertices contained in the same special structure. For every vertex v of G, let d(v) denote the degree of v in G, that is the number of hyperedges containing v.

Lemma 9. Let α be a special stucture, with the notations of Figure 1. Consider a (0, 6)-coloring of α.

We can change the color of the x i 's, y i 's and u such that the v i 's have no more neighbors colored k than before, and for all i, if v i is colored k, then v i has a neighbor colored 0.

Proof. If all of the v i 's are colored 0, then there is noting to do. If they are all colored k, then we assign 0 to u. If one of the v i 's, say v 0 , is colored 0 and another one, say v 1 , is colored k, then u and x 0 are colored k and we assign 0 to y 0 . Moreover, if α is a special configuration and v 2 is colored k, then y 2 is colored k and we assign 0 to x 2 .

Lemma 10. For every vertex v in G, d(v) -d(v) 7.
Proof. Let v be a vertex that does not verify the lemma, i.e. such that d(v)d(v) 6. As v is an 8 + -vertex, d(v) 1. Let α be a special structure incident to v in G. We use the notations of Figure 1, with say v = v 0 . The graph G -x 0 is smaller than G, thus it admits a (0, 6)-coloring. Since G does not admit a (0, 6)-coloring, v 0 is colored k and y 0 is colored 0. By Lemma 9, we can assume that v has a neighbor colored 0 in each of its special structures distinct from α. If v 1 is colored 0, then y 0 is colored k, a contradiction. Thus v 1 is colored k. If α is a special face, or if v 2 is colored k, then we assign 0 to u. If α is a special configuration and v 2 is colored 0, then x 2 is colored k and we assign 0 to y 2 . In both cases, v has at least d(v) neighbors colored 0. Thus v has at most d(v) -d(v) 6 neighbors colored k and we can assign k to v, a contradiction.

Lemma 11. Every component of G has at least one vertex v such that d(v)d(v) 8.

Proof. Suppose the lemma is false, and let C be a component of G that does not verify the lemma. If C has only one vertex, then this vertex is an 8 +vertex, which verifies d(v) -d(v) 8. Therefore C has at least one hyperedge, which corresponds to a special structure α of G. By Lemma 10, every vertex of C verifies d(v) -d(v) = 7. We use the notations of Figure 1. The graph G -{x 0 , y 0 } is smaller than G, thus it admits a (0, 6)-coloring. Since G admits no (0, 6)-coloring, v 0 and v 1 are colored k. If α is a special configuration and v 2 is colored 0, then x 2 and y 1 are colored k and we can color y 2 and x 1 with 0. Otherwise, we can color u with 0. Note that v 0 and v 1 , as well as v 2 if it exists and is colored k, all have six neighbors colored k, and by Lemma 9, we can assume that they all have at least one neighbor colored 0 in each of their special structures besides α.

If one of the v i 's, say v 0 , has an additional neighbor colored 0, it verifies d(v 0 ) -d(v 0 ) 8, a contradiction. Thus, for every v i , either v i is colored 0 or v i has no neighbor colored 0 outside of its special structures and at most one neighbor colored 0 in each special structure besides α.

We uncolor u and all the x i 's and y i 's, and let H equal to G where u, the x i 's and the y i 's are removed. By symmetry, we only consider the vertex v 0 . The following procedure either assigns 0 to v 0 or ensures that v 0 has two neighbors colored 0 in one of its special structures:

• For each special structure β containing v 0 and completely contained in H, we use the notations of Figure 1, keeping the same vertex for v 0 , but changing the other ones for the vertices in β, and do the following:

-By Lemma 9, we can assume that every v i colored k has a neighbor colored 0 in each of its special structures that are completely contained in H. -Suppose that one of the 8 + -vertices of β distinct from v 0 , say v 1 , has two neighbors colored 0 in a special structure distinct from β or a neighbor colored 0 outside of its special structures. Since d(v 1 )d(v 1 ) = 7, v 1 has at most five neighbors colored k outside of β if β is a special face, and at most four neighbors colored k outside of β if β is a special configuration. We assign k to y 0 k and 0 to x 0 . If v 2 exists and is colored 0, then we assign 0 to y 2 , and otherwise we assign 0 to u. We end the procedure. -We uncolor the 7 --vertices of β and remove them from H.

-For every 8 + -vertex w = v 0 in β colored k, we apply the procedure with w instead of v 0 . Now w is colored 0 or has two neighbors colored 0 in the same special structure.

• Suppose d = 7. Note that if there are two adjacent 2-vertices in α, then these two vertices are not in a 5-face, otherwise there would be a cycle of length 6 in G. The face α has an initial charge of 3, gives at most 3 4 to its 7 --vertices that are adjacent to an 8 + -vertex in α and nothing to its other vertices. There can be at most four of these vertices. Therefore ω (α) 3 -4 • 3 4 = 0. • Suppose d = 8. Note that at most one pair of adjacent 2-vertices is in a 5-face, otherwise there would be a cycle of length 6 in G. The face α has an initial charge of 4, gives at most 7 8 to its 7 --vertices that are adjacent to an 8 + -vertex in α, and nothing to its other vertices. There can be at most five of these vertices, and at most two are given 7 8 , the other being given at most 3 4 . Therefore ω (α) 4 -2

• 7 8 -3 • 3 4 = 0. • Suppose d = 9
. Note that at most two pairs of adjacent 2-vertices are in a 5-face, otherwise there would be a cycle of length 6 in G. The face α has an initial charge of 5, gives at most 7 8 to its 7 --vertices that are adjacent to an 8 + -vertex in α, and nothing to its other vertices. There can be at most six of these vertices, at most four are given 7 8 , and the others are given at most 3 4 . Therefore ω (α) 5 -4

• 7 8 -2 • 3 4 = 0. • Suppose d 10.
The face α has an initial charge of d -4, gives at most 7 8 to its 7 --vertices that are adjacent to an 8 + -vertex in α, and nothing to its other vertices. There can be at most d -4 of these vertices, therefore ω (α) d -4 -(d -4) • 7 8 > 0.

By Euler's formula, since G is connected by Lemma 4 and has at least one vertex, n + f -m = 2. The initial weight of the graph is

v∈V (G) ω(v) + α∈F (G) ω(α) = v∈V (G) (d(v) -4) + α∈F (G) (d(α) -4) = v∈V (G) d(v) + α∈F (G) d(α) -4n -4f = 4m -4n -4f = -8 < 0.
Therefore the initial weight of the graph is negative, thus the final weight of the graph is negative. Since by Lemmas 12 and 13, the final weight of every face and every vertex is non-negative, we get a contradiction. This completes the proof of Theorem 1.

Proof of Theorem 2

Let k 3 be a fixed integer. Suppose that there exists a graph in C that is not (0, k)-colorable. We consider such a graph H k that is minimal according to

. By adapting the proofs of Lemmas 4, 5, and 6, we obtain that the minimum degree of H k is at least two and every (k + 1) --vertex in H k is adjacent to a (k + 2) + -vertex. Suppose for contradiction that H k contains no 2-vertex. We consider the discharging procedure such that the initial chage of every vertex is equal to its degree and every 5 + -vertex gives 1 3 to every adjacent 3-vertex. Then the final charge of a 3-vertex is at least 3 + . This implies that the maximum average degree of H k is at least 10 3 , which is a contradiction since H k is a planar graph with girth at least 5. Thus, H k contains a 2-vertex v adjacent to the vertices u 1 and u 5 .

By minimality of H k , H k -v is (0, k)-colorable, every (0, k)-coloring of H k -v is such that u 1 and u 5 get distinct colors, and the vertex in {u 1 , u 5 } that is colored k has exactly k neighbors that are colored k.

Consider the graph H k obtained from H k -v by adding three 2-vertices u 2 , u 3 , and u 4 which form a path u 1 u 2 u 3 u 4 u 5 . Notice that H k is (0, k)-colorable and that in every (0, k)-coloring of H k is such that u 3 is colored k and is adjacent to exactly one vertex colored k. It is easy to see that H k is in C.

We are ready to prove that deciding whether a graph in C is (0, k)-colorable is NP-complete. The reduction is from the NP-complete problem of deciding whether a planar graph with girth at least 9 is (0, 1)-colorable [START_REF] Esperet | A complexity dichotomy for the coloring of sparse graphs[END_REF]. Given in instance G of this problem, we construct a graph G ∈ C, as follows For every vertex v in G, we add k -1 copies of H k and we add an edge between v and the vertex u 3 of each these copies. Notice that G is in C since G is planar and every cyle of length at most 8 is contained in a copy of

H k which is in C. Notice that a (0, 1)-coloring of G can be extended to a (0, k)-coloring of G . Conversely, a (0, k)-coloring of G induces a (0, 1)-coloring of G. So G is (0, 1)-colorable if and only if G is (0, k)-colorable.
4. A graph in C that is not (0, 3)-colorable Consider the graph F x,y depicted in Figure 2. Suppose for contradiction that F x,y admits a (0, 3)-coloring such that all the neighbors of x and y are colored 0 (the white vertices in the picture). Then the neighbors of those white vertices are colored k. We consider the 8 big vertices. Each of them is colored k and is ajacent to two vertices colored k. For every pair of adjacent red vertices, at least one of them is colored k. Notice that every red vertex is adjacent to a big vertex. Since there are 9 pairs of adjacent red vertices, there exists a big vertex that is adjacent to at least two red vertices colored k. This big vertex is thus adjacent to four vertices colored k, which is a contradiction.

In the graph depicted in Figure 3, every dashed line represent a copy of F x,y such that the extremities are x and y. Suppose for contradiction that this (C 3 , C 4 , C 6 )-free planar graph admits a (0, 3)-coloring. Each of the two drawn edges has at least one extremity colored k. Thus, there exist two vertices u and v colored k that are linked by 7 copies of F x,y . Since at most 3 neighbors of u and at most 3 neighbors of v can be colored k, one these 7 copies of F x,y is such that all the neighbors of x and y are colored 0. This is contradiction proves Theorem 2.

Following the proof above, we see that if we remove the green parts in Figures 2 and 3, we obtain a planar graph with girth 7 that is not (0, 2)-colorable.

A graph with such properties is already known [START_REF] Montassier | Near-colorings: non-colorable graphs and NP-completeness[END_REF], but this new graph is smaller (184 vertices instead of 1304) and the proof of non-(0, 2)-colorability is simpler.

Figure 1 :

 1 Figure 1: A special face (left) and a special configuration (right).

1 3 = 10 3 , 2 d

 32 the final charge of a d-vertex with d k + 2 is at least d -d × 1 3 /3 2 ( k + 2)/3 10 3 , and the final charge of every remaining vertex is at least 4 > 10 3

-We add back to H the 7 --vertices of β. If v 0 is colored 0, then we give them color k if they are adjacent to a vertex colored 0 and we assign them 0 otherwise, and we end the procedure. If β is a special face and v 1 is colored k, or if β is a special configuration and v 1 and v 2 are colored k, then we color u and x 0 with 0, we color the other 2-vertices with k, and we end the procedure. Suppose β is a special configuration, either v 1 or v 2 , say v 1 , is colored k, and the other one is colored 0. We assign 0 to x 0 , x 1 , and y 2 , and k to u, y 0 , y 1 , and x 1 , and we end the procedure. Now all of the v i 's distinct from v 0 are colored 0. We color x 0 and y 2 (if it exists) with 0 and we color the other 7 -vertices in β with color k.

• Now in each special structure containing v 0 and completely contained in H, all of the 8 + -vertices distinct from v 0 is colored 0. We assign 0 to v 0 and k to all of the neighbors of v 0 .

Let us prove that the previous procedure terminates. It always calls itself iteratively on a graph with fewer vertices, thus the number of nested iterations is bounded by the order of the initial graph. Furthermore, each iteration of the procedure only does a bounded number of calls to the procedure (at most two). That proves that the procedure terminates.

In the end, if one of the v i 's is colored k, then it has at most five neighbors colored k outside of α if α is a special face, and at most four neighbors colored k outside of α if α is a special structure. If every v i is colored k, then color u with color 0 and the other 7 --vertex of α with color k. Otherwise, assign k to u, and do the following:

• If every v i is colored 0, then assign k to the x i 's and the y i 's.

• If α is a special face and one of the v i 's, say v 0 , is colored 0 while the other one is colored k, then assign k to x 0 and 0 to y 0 .

• If α is a special structure, then assign k to the y i 's, and for all i ∈ {0, 1, 2}, if v i is colored k, then assign 0 to x i , and if v i is colored 0 then assign k to x i .

In all cases, we get a (0, 6)-coloring of G, a contradiction.

For each component C of G, we choose a vertex v in C such that d(v)-d(v) 8 as the root of C. We then choose an orientation of the edges of G such that the only vertices with no incoming edges are the roots (for example do a breadth first search from the root of each component). For each 8 + -vertex v, v is said to sponsor all of the special faces that correspond to its outgoing edges in G.

We are now going to give some weight on the vertices and faces of the graph. Initially, for all d, every d-vertex has weight d -4, and every d-face has weight d -4. Thus every face and every 4 + -vertex has non-negative initial weight.

We apply the following discharging procedure. 6 1. Every 8 + -vertex gives weight 1 2 to each of its 7 --neighbors, to each special face it sponsors, and to the 3-vertex of each special configuration it sponsors. Additionally, for every edge vw where v and w are 8 + -vertices, v and w each give 1 4 to each of the faces containing the edge vw, and 1 4 more to the face containing vw if there is only one face containing vw. 2. For each 3 + -vertex v with degree at most 7 in G, v gives 1 2 to each of its 2-neighbors. Moreover, it gives 1 2 to each of its 5-faces where it is adjacent to two 8 + -vertices and where there are two 2-vertices.

Each face f gives 1

4 to its 3 + -vertices with degree at most 7 that are consecutive to an 8 + -vertex, for each time they appear consecutively to an 8 + -vertex in the boundary of f . 4. Each 5-face gives 1 4 to each of its 2-vertices with no 2-neighbor and 5 8 to its 2-vertices with a 2-neighbor. 5. Each 7 + -face gives 3 4 to each of its 2-vertices that belong to a 5-face and have no 2-neighbors, 7 8 to each of its 2-vertices that belong to a 5-face and have a 2-neighbor, 1 2 to each of its 2-vertices that do not belong to a 5-face and have no 2-neighbors, and 3 4 to each of its 2-vertices that do not belong to a 5-face and have a 2-neighbor.

Let ω be the initial weight distribution, and let ω be the final weight distribution, after the discharging procedure.

Lemma 12. Every vertex v verifies ω (v) 0.

Proof. Let v be a vertex of degree d. We have ω(v) = d -4.

• Suppose first that d 8. The vertex v gives 1 2 to each of its 7 --neighbors and two times 1 4 for each of its 8 + -neighbors in Step 1, for a total of

0. If v does not sponsor all of its special structures, then d -d(v) 7, and

• to the faces for a total of at most -If v is in a 5-face and adjacent to no other 2-vertex, then it receives 1 from its 3 + -neighbors in Steps 1 and 2, 1 4 from its 5-face in Step 4, and 3 4 from its other face in Step 5. -If v is not in a 5-face and is adjacent to another 2-vertex, then it receives 1 2 from its 8 + -neighbor in Step 1, and 3 2 from its faces in Step 5.

-If v is in a 5-face and adjacent to no other 2-vertex, then it receives 1 from its 3 + -neighbors in Steps 1 and 2, and 1 from its faces in Step 5.

In all cases, v receives 2 over the procedure, and thus ω (v) = 2-4+2 = 0.

Lemma 13. Every face α satisties ω (α) 0.

Proof. Let α be a vertex of degree d. We have ω(α) = d -4.

• Suppose d = 5. If α is a special face, then it receives 1 2 in Step 1 and gives The only remaining case is when α has, in this consecutive order, two 2-vertices, an 8 + -vertex, a 3 + -vertex with degree at most 7, and another 8 + -vertex. In this case, α receives In all cases, ω (α) 1 -1 = 0.