Multiple Coloring of Cone Graphs

doi:10.1137/070691486

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Title:	Multiple Coloring of Cone Graphs
Authors:	Pan, Zhishi;Zhu, Xuding
Contributors:	淡江大學數學學系
Keywords:	Multiple coloring;Cone graphs;Mycielski graphs;Fractional chromatic number;Kneser graphs
Date:	2010
Issue Date:	2011-09-07 11:54:53 (UTC+8)
Publisher:	Philadelphia: Society for Industrial and Applied Mathematics
Abstract:	A k-fold coloring of a graph assigns to each vertex a set of k colors, and color sets assigned to adjacent vertices are disjoint. The kth chromatic number Xk(G) of a graph G is the minimum total number of colors needed in a k-fold coloring of G. Given a graph G = (V, E) and an integer m ≥ 0, the m-cone of G, denoted by µm(G), has vertex set (V x {0,1,… , m}) U {u} in which u is adjacent to every vertex of V x {m}, and (x, i)(y, j) is an edge if xy ∈ E and i = j = 0 or xy ∈ E and \|i - j\| = 1. This paper studies the kth chromatic number of the cone graphs. An upper bound for Xk(µm(G) in terms of Xk(G), k, and m are given. In particular, it is proved that for any graph G, if m ≥ 2k, then Xk(µm(G)) ≤ Xk(G) + 1. We also find a surprising connection between the kth chromatic number of the cone graph of G and the circular chromatic number of G. It is proved that if Xk(G)/k > Xc((G) and Xk(G) is even, then for sufficiently large m, Xk(µm(G)) = Xk(G). In particular, if X(G) > Xc(G) and X(G) is even, then for sufficiently large m, X(µm(G)) = X(G).
Relation:	SIAM Journal on Discrete Mathematics 24(4), pp.1515-1526
DOI:	10.1137/070691486
Appears in Collections:	[Graduate Institute & Department of Mathematics] Journal Article

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