List-coloring graphs without subdivisions and without immersions

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 listcoloring 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 Hájos’ conjecture and Hadwiger’s conjecture. We point out that it is Unique-Game hard to obtain an O(k/ log 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/ log 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 graphcoloring of H-minor-free graphs [13]). ∗National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, Japan. Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by C & C Foundation, by Kayamori Foundation and by Inoue Research Award for Young Scientists. Email: k [email protected] †University of Tokyo, Tokyo 113-8656, Japan. Supported by Grant-in-Aid for Scientific Research and by the Global COE Program “The research and training center for new development in mathematics” from Ministry of Education, Culture, Sports, Science and Technology of Japan. E-mail: [email protected] We also show that the Erdős-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 edgeconnectivity condition is necessary in this statement.