List-chromatic Number and the Chromatic Number in Minor-closed and Odd-minor-closed Classes of Graphs
نویسندگان
چکیده
It is well-known (Feige and Kilian [24], H̊astad [39]) that approximating the chromatic number within a factor of n1−ε cannot be done in polynomial time for ε > 0, unless coRP = NP. Computing the list-chromatic number is much harder than determining the chromatic number. It is known that the problem of deciding if the list-chromatic number is k, where k ≥ 3, is Πp2-complete [37]. In this paper, we focus on minor-closed and odd-minor-closed families of graphs. In doing that, we may as well consider only graphs without Kk-minors and graphs without odd Kk-minors for a fixed value of k, respectively. Our main results are that there is a polynomial time approximation algorithm for the list-chromatic number of graphs without Kk-minors and there is a polynomial time approximation algorithm for the chromatic number of graphs without odd-Kk-minors. Their time complexity is O(n) and O(n), respectively. The algorithms have multiplicative error O( √ log k) and additive error O(k), and the multiplicative ∗Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by Sumitomo Foundation, C & C Foundation and by Inoue Research Award for Young Scientists. †Current address:National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430 ‡Supported in part by the Ministry of Science and Technology of Slovenia, Research Program P1–0297.
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