Colouring graphs with bounded generalized colouring number

Abstract

AbstractGiven a graph G and a positive integer p, χp(G) is the minimum number of colours needed to colour the vertices of G so that for any i≤p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χp(G) in terms of the k-colouring number colk(G) of G for k=2p−2. Conversely, for each integer k, we also prove an upper bound for colk(G) in terms of χk+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent: (a)For every positive integer p, χp(G) is bounded by a constant for all G∈K.(b)For every positive integer k, colk(G) is bounded by a constant for all G∈K. It was proved by Nešetřil and Ossona de Mendez that (a) is equivalent to the following: (c)For every positive integer q, ∇q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all G∈K. Therefore (b) and (c) are also equivalent. We shall give a direct proof of this equivalence, by introducing ∇q−(1/2)(G) and by showing that there is a function Fk such that ∇(k−1)/2(G)≤(colk(G))k≤Fk(∇(k−1)/2(G)). This gives an alternate proof of the equivalence of (a) and (c)