Colouring graphs with bounded generalized colouring number

Abstract

Given a graph G and a positive integer p , χ p ( 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 χ p ( G ) in terms of the k -colouring number col k ( G ) of G for k = 2 p − 2 . Conversely, for each integer k , we also prove an upper bound for col k ( G ) in terms of χ k + 2 ( G ) . As a consequence, for a class K of graphs, the following two statements are equivalent: (a) For every positive integer p , χ p ( G ) is bounded by a constant for all G ∈ K . (b) For every positive integer k , col k ( 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 F k such that ∇ ( k − 1 ) / 2 ( G ) ≤ ( col k ( G ) ) k ≤ F k ( ∇ ( k − 1 ) / 2 ( G ) ) . This gives an alternate proof of the equivalence of (a) and (c).