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On the Strong Chromatic Number of Random Graphs

Published online by Cambridge University Press:  01 March 2008

PO-SHEN LOH  and
BENNY SUDAKOV
PO-SHEN LOH
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA (e-mail: [email protected])
BENNY SUDAKOV
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA and Institute for Advanced Study, Princeton, USA (e-mail: [email protected])
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Abstract

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colourable if, for every partition of V(G) into disjoint sets V1 ∪ ··· ∪ Vr, all of size exactly k, there exists a proper vertex k-colouring of G with each colour appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colourable if the graph obtained by adding isolated vertices is strongly k-colourable. The strong chromatic number of G is the minimum k for which G is strongly k-colourable. In this paper, we study the behaviour of this parameter for the random graph Gn,p. In the dense case when p ≫ n−1/3, we prove that the strong chromatic number is a.s. concentrated on one value Δ + 1, where Δ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 2 , March 2008 , pp. 271 - 286
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Aharoni, R., Berger, E. and Ziv, R. Independent systems of representatives in weighted graphs. Combinatorica, to appear.Google Scholar
[2]Aharoni, R. and Holzman, R. (2005) Private communication.Google Scholar
[3]Alon, N. (1988) The linear arboricity of graphs. Israel J. Math. 62 311–325.CrossRefGoogle Scholar
[4]Alon, N. (1992) The strong chromatic number of a graph. Random Struct. Alg. 3 1–7.CrossRefGoogle Scholar
[5]Alon, N. (2003) Problems and results in extremal combinatorics I. Discrete Math. 273 31–53.CrossRefGoogle Scholar
[6]Alon, N. and Krivelevich, M. (1997) The concentration of the chromatic number of random graphs. Combinatorica 17 303–313.CrossRefGoogle Scholar
[7]Alon, N. and Spencer, J. (2000) The Probabilistic Method, 2nd edn, Wiley, New York.CrossRefGoogle Scholar
[8]Bohman, T. and Holzman, R. (2002) On a list coloring conjecture of Reed. J. Graph Theory 41 106–109.CrossRefGoogle Scholar
[9]Bollobás, B. (2001) Random Graphs, 2nd edn, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[10]Bollobás, B., Erdős, P. and Szemerédi, E. (1975) On complete subgraphs of r-chromatic graphs. Discrete Math. 13 97–107.CrossRefGoogle Scholar
[11]Fellows, M. (1990) Transversals of vertex partitions in graphs. SIAM J. Discrete Math. 3 206–215.CrossRefGoogle Scholar
[12]Fleischner, H. and Stiebitz, M. (1992) A solution to a colouring problem of P. Erdős. Discrete Math. 101 39–48.CrossRefGoogle Scholar
[13]Fleischner, H. and Stiebitz, M. (1997) Some remarks on the cycle plus triangles problem. In The Mathematics of Paul Erdős, Vol. II, Algorithms and Combinatorics 14 136–142.CrossRefGoogle Scholar
[14]Haxell, P. (2001) A note on vertex list colouring. Combin. Probab. Comput. 10 345–348.CrossRefGoogle Scholar
[15]Haxell, P. (2004) On the strong chromatic number. Combin. Probab. Comput. 13 857–865.CrossRefGoogle Scholar
[16]Haxell, P. (2005) Private communication.Google Scholar
[17]Haxell, P. and Szabó, T. (2006) Odd transversals are odd. Combin. Probab. Comput. 15 193–211.CrossRefGoogle Scholar
[18]Jin, G. (1992) Complete subgraphs of r-partite graphs. Combin. Probab. Comput. 1 241–250.CrossRefGoogle Scholar
[19]Loh, P. and Sudakov, B. Independent transversals in locally sparse graphs. J. Combin. Theory Ser. B, to appear.Google Scholar
[20]Łuczak, T. (1991) A note on the sharp concentration of the chromatic number of random graphs. Combinatorica 11 295–297.CrossRefGoogle Scholar
[21]Matula, D. (1970) On the complete subgraphs of a random graph. In Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications, University of North Carolina, Chapel Hill, NC, pp. 356–369.Google Scholar
[22]Meshulam, R. (2003) Domination numbers and homology. J. Combin. Theory Ser. A 102 321–330.CrossRefGoogle Scholar
[23]Reed, B. (1999) The list colouring constants. J. Graph Theory 31 149–153.3.0.CO;2-#>CrossRefGoogle Scholar
[24]Reed, B. and Sudakov, B. (2002) Asymptotically the list colouring constants are 1. J. Combin. Theory Ser. B 86 27–37.CrossRefGoogle Scholar
[25]Shamir, E. and Spencer, J. (1987) Sharp concentration of the chromatic number on random graphs G n,p. Combinatorica 7 121–129.CrossRefGoogle Scholar
[26]Szabó, T. and Tardos, G. (2006) Extremal problems for transversals in graphs with bounded degree. Combinatorica 26 333–351.CrossRefGoogle Scholar
[27]Yuster, R. (1997) Independent transversals in r-partite graphs. Discrete Math. 176 255–261.CrossRefGoogle Scholar