On the strong chromatic number of random graphs

Authors:
Po-shen Loh;Benny Sudakov
Affiliations:
Department of mathematics, princeton university, princeton, nj 08544, usa (e-mail: ploh@math.princeton.edu);Department of mathematics, princeton university, princeton, nj 08544, usa and institute for advanced study, princeton, usa (e-mail: bsudakov@math.princeton.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2008

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Abstract

Let G be a graph with n vertices, and let kbe an integer dividing n. G is said to be stronglyk-colourable if, for every partition of V(G) intodisjoint sets V1∪ ···∪ Vr, all of size exactly k,there exists a proper vertex k-colouring of G witheach colour appearing exactly once in eachVi. In the case when k does notdivide n, G is defined to be stronglyk-colourable if the graph obtained by addingk[n/k]-n isolated vertices is stronglyk-colourable. The strong chromatic number of G is theminimum k for which G is stronglyk-colourable. In this paper, we study the behaviour of thisparameter for the random graphGn,p. In the dense case whenp n1/3, we prove that the strongchromatic number is a.s. concentrated on one value Δ + 1,where Δ is the maximum degree of the graph. We also obtainseveral weaker results for sparse random graphs.