List-coloring graphs without subdivisions and without immersions

Abstract

A graph G contains a subdivision of H if G contains a subgraph which is isomorphic to a graph that can be obtained from H by subdividing some edges. A graph H is immersed in a graph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. Although the well-known Kuratowski's theorem can be stated in terms of both a subdivision and a minor, we know that the notions of a subdivision and a minor do not seem to be similar. The notions of an immersion and a minor seem to be quite similar, and structural approach concerning graph minors has been extremely successful. In fact, Robertson and Seymour extended their proof of the famous Wanger's conjecture to prove that graphs are well-quasi-ordered by the immersion relation. We give additive approximation algorithms for list-coloring within 3.5(k + 1) of the list-chromatic number for graphs without Kk as a subdivision, and within 1.5(k - 1) of the list-chromatic number for graphs without Kk as an immersion. Clearly our results give rise to additive approximation algorithms for graph-coloring of graphs without Kk as a subdivision (in fact, we shall give an additive approximation algorithm within 2.5(k + 1) of the chromatic number) and Kk as an immersion, too. These are the first results in this direction (in fact, these are the first results concerning list-coloring graphs without fixed graph as a subdivision or as an immersion, except for the known upper bound results) and extend the result by Kawarabayashi, Demaine and Hajiaghayi (SODA'09) concerning the additive approximation algorithm for list-coloring graphs without Kk as a minor. We also discuss how our results are related to the famous Hajos' conjecture and Hadwiger's conjecture. We point out that it is Unique-Game hard to obtain an O(k/ log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k - 2 [6], and hence it is also Unique-Game hard to obtain an O(k/ log2 k)-approximation algorithm for graph-coloring of graphs without a Kk-subdivision or without a Kk-immersion. Therefore it really makes sense to consider an additive approximation algorithm for graph coloring of these family of graphs (which is in contrast to a 2-approximation algorithm for graph-coloring of H-minor-free graphs [13]). We also show that the Erdocombining double acute accents-Pósa-property holds for a Kt-immersion in 1.5t-edge-connected graph G. More precisely, there exists a function f(k, t) such that either G has k edge-disjoint Kt-immersions, or G has an edge set S of order at most f(k, t) such that G - S has no Kt- immersion. We finally construct an example that shows that some edge-connectivity condition is necessary in this statement. Copyright © SIAM.