Multiple Coloring of Cone Graphs

The kth chromatic number χk(G) of a graph G is the minimum number of colours needed so that each vertex can be assigned a set of k colours in such a way that colour sets assigned to adjacent vertices are disjoint. Given a graph G = (V,E) and an integerm ≥ 0, them-cone ofG, denoted by μm(G), has vertex set (V ×{0, 1, · · · ,m})∪{u} in which u is adjacent to every vertex of V ×{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 χk(μm(G)) in terms of χk(G), k and m is given. In particular, it is proved that for any graph G, if m ≥ 2k, then χk(μm(G)) ≤ χk(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 χk(G)/k > χc(G) and χk(G) is even, then for sufficiently large m, χk(μm(G)) = χk(G).