Colouring graphs with bounded generalized colouring number
Colouring graphs with bounded generalized colouring number
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DOI：
10.1016/j.disc.2008.03.024
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发表时间：
2009-09
期刊：
影响因子：
--
通讯作者：
Xuding Zhu
中科院分区：
文献类型：
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作者：
Xuding Zhu
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: It was proved by Nešetřil and Ossona de Mendez that (a) is equivalent to the following: 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 Fksuch that [Formula: see text] . This gives an alternate proof of the equivalence of (a) and (c).
