Colouring graphs with bounded generalized colouring number

Given a graph G and a positive integer p, @g"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 @g"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 @g"k"+"2(G). As a consequence, for a class K of graphs, the following two statements are equivalent: (a)For every positive integer p, @g"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 Nesetril 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).