Approximating List-Coloring on a Fixed Surface

Abstract

It is well-known that approximating the chromatic number within a factor of n 1 − ε cannot be done in polynomial time for any ε> 0, unless coRP = NP. Also, it is known that computing the list-chromatic number is much harder than the chromatic number (assuming that the complexity classes NP and coNP are different). In fact, the problem of deciding if a given graph is f-list-colorable for a function f : V →{k − 1,k} for k ≥ 3 is Π 2 p-complete.In this paper, we are concerned with the following questions: 1 Given a graph embedded on a surface of bounded genus, what is its list-chromatic number ? 1 Given a graph embedded on a surface of bounded genus with list-chromatic number k, what is the least l (l ≥ k) such that the graph can be efficiently and legally colored given a list (coloring scheme) of order l ? The seminal result of Thomassen [19] gives rise to answers for these problems when a given graph is planar. In fact, he gave a polynomial time algorithm to 5-list-color a planar graph. Thomassen’s result together with the hardness result (distinguishing between 3, 4 and 5 list-colorability is NP-complete for planar graphs and bounded genus graphs) gives an additive approximation algorithm for list-coloring planar graphs within 2 of the list-chromatic number.Our main result is to extend this result to bounded genus graphs. In fact, our algorithm gives a list-coloring when each vertex has a list with at least χ l (G) + 2 colors available. The time complexity is O(n).It also generalizes the other deep result of Thomassen [20] who gave an additive approximation algorithm for graph-coloring bounded genus graphs within 2 of the chromatic number.This theorem can be compared to the result by Kawarabayashi and Mohar(STOC’06) who gave an O(k)-approximation algorithm for list-coloring graphs with no K k -minors. For minor-closed graphs, there is a 2-approximation algorithm for graph-coloring by Demaine, Hajiaghayi and Kawarabayashi (FOCS’05), but it seems that there is a huge gap between list-coloring and graph-coloring in minor-closed family of graphs. On the other hand, this is not the case for bounded genus graphs, as we pointed out above.KeywordsPlanar GraphChromatic NumberBlock DecompositionNest CycleColor TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.