. Proof, Il existe bien un moyen de subdiviser les packings selon la proposition 4, p.31

. Proof, Il existe bien un moyen de subdiviser les packings selon la proposition 4

. Cependant, 2 t ) [86] pour déterminer si ?(G) ? t, pour un graphe G. Ce problème est donc dans la classe XP avec comme paramètre t. Notre démarche est motivée par une série d'articles qui concernent le nombre b-chromatique des graphes réguliers

G. Cette-remarque-a-une-conséquence-un-graphe, importe quelle t-coloration de Grundy partielle, il existe des plus petits sousgraphes H de G tels que ?(H) ? t. La famille des t-atomes comprend tous ces plus petits sous-graphes. Ce concept à été introduit par Zaker [86]. La famille des t-atomes est nie et la présence d'un t-atome peut être déterminée en un temps polynomial, pour un entier t xé, La dénition suivante est quelque peu diérente de la dénition originale de Zaker. Cette dénition insiste plus sur la construction de tous les t-atomes (et non pas de quelques t-atomes comme pour Zaker)

C. Soit, ordre 3 ou 4 qui contient v ou un voisin de v et soit D 1 = {x ? V (G)| d(x, C) = 1}, où d(x, C) est la distance de x à C dans le graphe G

N. De and G. Entrée, Un graphe G. Paramètre: Le degré maximum ?(G) Question: Déterminer ?(G)

. Proof, Pour cela il sut de déterminer tous les sous-ensembles d'au plus |S| sommets dans N (S)

. Vérier-qu-'un-sous-ensemble-est-un-stable and . Qu, un sous-ensemble est minimal pour la domination de S se fait en temps linéaire, On a donc un algorithme qui fonctionne en temps O((?(G)|S|) |S| )

. Szemerédi, on introduit un problème d'optimisation en vue d'organiser les grand graphes. Ce problème d'optimisation est une alternative à la minimisation du nombre d'arêtes coupées entre les clusters (le min-cut) Un article a été soumis à ce sujet

]. I. Alegre, M. A. Fiol, and J. L. Yebra, Some large graphs with given degree and diameter, Journal of Graph Theory, vol.33, issue.2, p.219224, 1986.
DOI : 10.1002/jgt.3190100211

S. Antonucci, Generalizzazioni del concetto di cromatismo d'un grafo, Boll. Un. Mat. Ital, vol.15, p.2031, 1978.

G. Argiroo, Polynomial instances of the packing coloring problem, Electronic Notes in Discrete Mathematics, p.363368, 2011.

G. Argiroo, G. Nasini, and P. Torres, The packing coloring problem for (q,q-4) graphs, Lecture Notes in Computer Science, vol.7422, p.309319, 2012.

S. Arora and B. Barak, Computational Complexity : A modern Approach, 2009.
DOI : 10.1017/CBO9780511804090

M. Asté, F. Havet, and C. L. Sales, Grundy number and products of graphs, Discrete Mathematics, vol.310, issue.9, p.14821490, 2010.
DOI : 10.1016/j.disc.2009.09.020

C. Berge, Théorie des graphes et ses applications, Dunod, 1958.

D. Bienstock, N. Robertson, P. Seymour, and R. Thomas, Quickly excluding a forest, Journal of Combinatorial Theory, Series B, vol.52, issue.2, p.274283, 1991.
DOI : 10.1016/0095-8956(91)90068-U

M. Blidiaa, F. Maray, and Z. Zemira, On b-chromatic number in regular graphs, Discrete Applied Mathematics, vol.157, p.17871793, 2009.

M. Bonamy and N. Bousquet, Brooks??? theorem on powers of graphs, Discrete Mathematics, vol.325, 2013.
DOI : 10.1016/j.disc.2014.01.024
URL : https://hal.archives-ouvertes.fr/lirmm-01264422

M. Bonamy, B. Lévêque, and A. Pinlou, List coloring the square of sparse graphs with large degree, European Journal of Combinatorics, vol.41, 2013.
DOI : 10.1016/j.ejc.2014.03.006
URL : https://hal.archives-ouvertes.fr/lirmm-01233445

O. Borodin, On the total coloring of planar graphs, J. Reine Angew. Math, vol.394, p.180185, 1989.

B. Bresar, S. Klavzar, and D. F. , On the packing chromatic number of Cartesian products, hexagonal lattice, and trees, 15] R. L. Brooks. On colouring the nodes of a network. Proc. Cambridge Philosophical Society, pp.2303-2311, 1941.
DOI : 10.1016/j.dam.2007.06.008

S. Cabello and M. Jakovac, On the b-chromatic number of regular graphs, Discrete Applied Mathematics, vol.159, issue.13, p.13031310, 2011.
DOI : 10.1016/j.dam.2011.04.028

S. Chiang and J. Yan, On L(d, 1)-labeling of cartesian product of a cycle and a path, Discrete Applied Mathematics, vol.156, p.28672881, 2008.

M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Annals of Mathematics, vol.164, issue.1, p.51229, 2006.
DOI : 10.4007/annals.2006.164.51

C. V. Conta, Torus and Other Networks as Communication Networks With Up to Some Hundred Points, IEEE Transactions on Computers, vol.32, issue.7, p.657666, 1983.
DOI : 10.1109/TC.1983.1676297

S. Cook, The complexity of theorem-proving procedures, Proceedings of the third annual ACM symposium on Theory of computing , STOC '71, p.151158, 1971.
DOI : 10.1145/800157.805047

B. Courcelle, The monadic second-order logic of graphs i. recognizable sets of nite graphs, Information and Computation, vol.85, p.1275, 1990.

D. Cranston and S. Kim, List-coloring the square of a subcubic graph, Journal of Graph Theory, vol.17, issue.1, p.6587, 2008.
DOI : 10.1002/jgt.20273

B. Darties, N. Gastineau, and O. Togni, Completely independent spanning trees in cartesian product of cycle and clique

N. G. De-bruijn and P. Erd®s, A colour problem for innite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A, vol.54, p.371373, 1951.

Z. Dvo°áka, D. Král, P. Nejedlýa, and R. ’krekovski, Coloring squares of planar graphs with girth six, European Journal of Combinatorics, vol.29, pp.838-849, 2008.

B. Eantin and H. Kheddouci, Grundy number of graphs, Discussiones Mathematicae Graph theory, vol.27, p.518, 2007.

J. Ekstein, J. Fiala, P. Holub, and B. Lidický, The packing chromatic number of the square lattice is at least 12, 2010.

J. Ekstein, P. Holub, and B. Lidický, Packing chromatic number of distance graphs, Discrete Applied Mathematics, vol.160, issue.4-5, p.518524, 2012.
DOI : 10.1016/j.dam.2011.11.022

J. Ekstein, P. Holub, and O. Togni, The packing coloring of distance graphs D(k, t), Discrete Applied Mathematics, vol.167, p.100106, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00905732

P. Erd®s, S. T. Hedetniemi, R. C. Laskar, and G. C. Prins, On the equality of the partial Grundy and upper ochromatic number of graphs, Discrete Mathematics, vol.272, p.5364, 2003.

A. ’evcková, Distant chromatic number of the planar graphs, 2001.

G. Fertin, E. Godard, and A. Raspaud, Acyclic and k-distance coloring of the grid, Information Processing Letters, vol.87, issue.1, p.5158, 2003.
DOI : 10.1016/S0020-0190(03)00232-1
URL : https://hal.archives-ouvertes.fr/hal-00307787

J. Fiala and P. A. Golovach, Complexity of the packing coloring problem for trees, Discrete Applied Mathematics, vol.158, p.771778, 2010.

J. Fiala, S. Klavºar, and B. Lidický, The packing chromatic number of innite product graphs, Europan Journal of Combinatorics, vol.30, p.11011113, 2009.

A. S. Finbow and D. F. , On the packing chromatic number of some lattices, Discrete Applied Mathematics, vol.158, issue.12, p.12241228, 2010.
DOI : 10.1016/j.dam.2009.06.001

F. Fu, S. Ling, and C. Xing, New results on two hypercube coloring problems, Discrete Applied Mathematics, vol.161, issue.18, p.29372945, 2013.
DOI : 10.1016/j.dam.2013.07.006

N. Gastineau, On dichotomies among the instance of the S-coloring problem

N. Gastineau, Complexity of the (s 1 , s 2 , . . . , s k )-packing coloring problem, Bordeaux Graph Workshop, 2012.

N. Gastineau and H. Kheddoucci, A clustering organisation based on the -regularity

N. Gastineau, H. Kheddouci, and O. Togni, Subdivision into i-packings and s-packing chromatic number of some lattices
URL : https://hal.archives-ouvertes.fr/hal-01157901

N. Gastineau, H. Kheddouci, and O. Togni, On the family of -regular graphs with Grundy number, Discrete Mathematics, vol.328, p.515, 2014.
DOI : 10.1016/j.disc.2014.03.023
URL : https://hal.archives-ouvertes.fr/hal-00922022

N. Gastineau and O. Togni, S-packing colorings of cubic graphs
URL : https://hal.archives-ouvertes.fr/hal-00967446

M. Gionfriddo, Sulle colorazioni L s d'un grafo nito, pp.15-444454, 1978.

M. Gionfriddo, Sul parametro ? ? (g) d'un grafo ls-colorabile e problemi relativi, Riv. Mat. Univ. Parma, vol.8, p.17, 1982.

W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, J. M. Harris, and D. F. , Broadcast chromatic numbers of graphs, Ars Combinatoria, vol.86, p.3349, 2008.

W. Goddard and H. Xu, The S-packing chromatic number of a graph Discussiones Mathematicae Graph Theory, Xu. A note on S-packing colorings of lattices, p.795806255262, 2012.

P. M. Grundy, Mathematics and games, Eureka, vol.2, p.68, 1939.

F. Havet and L. Sampaio, On the Grundy and b-Chromatic Numbers of a Graph, Algorithmica, vol.306, issue.23, p.885899, 2013.
DOI : 10.1007/s00453-011-9604-4
URL : https://hal.archives-ouvertes.fr/hal-00905927

J. Heuvel and S. Mcguinnes, Colouring the square of a planar graph, Journal of Graph Theory, vol.42, p.110124, 2003.

A. J. Homan and R. R. Singleton, On Moore graphs with diameters 2 and 3, IBM Journal of Research and Development, vol.5, p.497504, 1960.

P. Jacko and S. Jendrol, Distance Coloring of the Hexagonal Lattice, Discussiones Mathematicae Graph Theory, vol.25, issue.1-2, p.151166, 2007.
DOI : 10.7151/dmgt.1269

M. Jakovac and S. Klavºar, The b-Chromatic Number of Cubic Graphs, Graphs and Combinatorics, vol.306, issue.1, p.107118, 2010.
DOI : 10.1007/s00373-010-0898-9

S. Jendrol and Z. Skupien, Local structures in plane maps and distance colourings, Discrete Mathematics, vol.236, issue.1-3, p.167144, 2001.
DOI : 10.1016/S0012-365X(00)00440-4

J. Kim and S. Kim, The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes, Discrete Applied Mathematics, vol.159, issue.18, pp.2222-2228, 2011.
DOI : 10.1016/j.dam.2011.07.022

D. Korºe and A. Vesel, On the packing chromatic number of square and hexagonal lattice, Ars mathmematica contemporanea, vol.7, p.1322, 2014.

F. Kramer and H. Kramer, Un probleme de coloration des sommets d'un graphe, C.R. Acad. Sci. Paris A, vol.268, p.4648, 1969.

F. Kramer and H. Kramer, On the Generalized Chromatic Number, Ann. Discrete Math, vol.30, p.275284, 1986.
DOI : 10.1016/S0304-0208(08)73145-1

F. Kramer and H. Kramer, A survey on the distance-colouring of graphs. Discrete mathematics, p.422426, 2008.

K. Kuratowski, Sur le problème des courbes gauches en topologie, Fund. Math, vol.15, p.271283, 1930.

L. Levin, Universal search problems. Problems of Information Transmission, p.115116, 1973.

K. Lih, W. Wang, and X. Zhu, Coloring the square of a K4-minor free graph, Discrete Mathematics, vol.269, issue.1-3, p.303309, 2003.
DOI : 10.1016/S0012-365X(03)00059-1

M. S. Molloy, Frequency Channel Assignment on Planar Networks, Lecture Notes in Computer Science, p.736747, 2002.
DOI : 10.1007/3-540-45749-6_64

E. S. Mahmoodian and F. S. Mousavi, Coloring the square of products of cycles and paths, J. Combin. Math. Combin. Comput, vol.76, p.101119, 2011.

L. Miao and Y. Fan, The distance coloring of graphs, Acta Mathematica Sinica, English Series, vol.42, issue.2, 2012.
DOI : 10.1007/s10114-014-3238-9

C. Papadimitriou, Computational Complexity, 1993.

G. Royle, List of cubic graphs

A. E. Sahili and M. Kouidder, b-chromatic number of regular graphs, Utilitas Mathematica, vol.80, p.211216, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01301071

A. E. Sahili, M. Kouidder, and M. Maidoun, b-chromatic number of regular graphs, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01301071

L. Sampaio, Algorithmic aspects of graph colouring heuristics, 2012.

S. Shaebani, On the b-chromatic number of regular graphs without 4-cycle, Discrete Applied Mathematics, vol.160, p.16101614, 2012.

Z. Shao and A. Vesel, A note on the chromatic number of the square of the Cartesian product of two cycles, Discrete Mathematics, vol.313, issue.9, p.9991001, 2013.
DOI : 10.1016/j.disc.2013.01.025

A. Sharp, Distance coloring, ESA, p.510521, 2007.
DOI : 10.1007/978-3-540-75520-3_46

Z. Shi, W. Goddard, S. T. Hedetniemi, K. Kennedy, R. Laskar et al., An algorithm for partial Grundy number on trees, Discrete Mathematics, vol.304, issue.1-3, p.108116, 2005.
DOI : 10.1016/j.disc.2005.09.008

C. Sloper, An eccentric coloring of trees, Australasian Journal of Combinatorics, vol.29, p.309321, 2004.

E. Sopena and J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Mathematics, vol.310, issue.17-18, p.23272333, 2010.
DOI : 10.1016/j.disc.2010.05.011
URL : https://hal.archives-ouvertes.fr/hal-00392145

R. Soukal and P. Holub, A note on packing chromatic number of the square lattice, The Electronic Journal of Combinatorics, vol.17, p.447468, 2010.

O. Togni, On packing colorings of distance graphs, Discrete Applied Mathematics, vol.167, p.280289, 2014.
DOI : 10.1016/j.dam.2013.10.026
URL : https://hal.archives-ouvertes.fr/hal-00531583

P. Torres, On the packing chromatic number of hypercubes, Electronic Notes in Discrete Mathematics, vol.44, p.263268, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00926875

C. Wegner, A. William, and S. Roy, Graphs with given diameter and a colouring problem Packing chromatic number of certain graphs, International Journal of Pure and Applied Mathematics, vol.87, p.731739, 1977.

S. Wong, Colouring graphs with respect to distance, 1996.

S. Yahiaoui and B. Eantin, On the grundy number of cubic graphs. preprint, 2012.

M. Zaker, Results on the Grundy chromatic number of graphs, Discrete Mathematics, vol.306, issue.23, p.31663173, 2006.
DOI : 10.1016/j.disc.2005.06.044

. Dans-ce-mémoire, nous regroupons tous les résultats connus à propos de la S -coloration de packing