List-color-critical graphs on a fixed surface

A k-list-assignment for a graph G assigns to each vertex v of G a list L(v) of admissible colors , where |L(v)| ≥ k. A graph is k-list-colorable (or k-choosable) if it can be properly colored from the lists for every k-listassignment. We prove the following conjecture posed by Thomassen in 1994: “There are only finitely many listcolor-critical graphs with all lists of cardinality at least 5 on any fixed surface.” This generalizes the well-known result of Thomassen on the usual graph coloring case. We use this theorem and specific parts of its proof to resolve the complexity status of the following problem about k-list-coloring graphs on a fixed surface S, where k is a fixed positive integer. Input: A graph G embedded in the surface S. Question: Is G k-choosable? If not, provide a certificate (a list-color-critical subgraph and the corresponding k-list-assignment). The cases k = 3, 4 are known to be NP-hard (actually even Πp2-complete), and the cases k = 1, 2 are easy. Our main results imply that the problem is tractable for every k ≥ 5. In fact, together with our recent algorithmic result, we are able to solve it in linear time when k ≥ 5. Our proof yields even more: if the input graph is k-list-colorable, then for any k-listassignment L, we can construct an L-coloring of G in linear time. This generalizes the well-known linear-time algorithms for planar graphs by Nishizeki and Chiba (for 5-coloring), and Thomassen (for 5-list-coloring). We also give a polynomial-time algorithm to resolve the following question: Input: A graph G in the surface S, and a k-listassignment L, where k ≥ 5. Question: Does G admit an L-coloring? If not, ∗Research partly supported by JSPS Postdoctoral Fellowship for Research Abroad. †National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan ‡E-mail: k [email protected] §Supported in part by an NSERC Discovery Grant, by an NSERC Accelerator Supplement, by the CRC Program, and by the Ministry of Higher Education, Science and Technology of Slovenia, Research Program P1-0297. ¶Department of Mathematics, Simon Fraser University, Burnaby, B.C. V5A 1S6, Canada ‖email: [email protected] provide a certificate for this. If yes, then return an Lcoloring. If the graph G is k-list-colorable, then our first result gives a linear time solution. However, the second problem is more general, since it provides a coloring (or a small obstruction) for an arbitrary graph in S. We also use our main theorem to prove another conjecture that was proposed recently by Thomassen: “For every fixed surface S, there exists a positive constant c such that every 5-list-colorable graph with n vertices embedded on S, has at least c ·2n distinct 5-listcolorings for every 5-list-assignment for G.” Thomassen himself proved that this conjecture holds for usual 5colorings. In addition to all these results, we also made partial progress towards a conjecture of Albertson concerning coloring extensions and a progress on similar questions for triangle-free graphs and graphs of larger girth.