Colouring graphs with bounded generalized colouring number

Given 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 grad, i.e., the greatest reduced average density, of G with rank q) is bounded by a constant for all G ∈ K. This implies that (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)) ≤ Fk(∇(k−1)/2(G)). This gives an alternate proof of the equivalence of (a) and (c).