List colourings of planar graphs
Discrete Mathematics
Restricted colorings of graphs
Surveys in combinatorics, 1993
Graph minors. XI.: circuits on a surface
Journal of Combinatorial Theory Series B
Every planar graph is 5-choosable
Journal of Combinatorial Theory Series B
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Graph minors. XII: distance on a surface
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Color-critical graphs on a fixed surface
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Highly connected sets and the excluded grid theorem
Journal of Combinatorial Theory Series B
The extremal function for complete minors
Journal of Combinatorial Theory Series B
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
Approximation algorithms for classes of graphs excluding single-crossing graphs as minors
Journal of Computer and System Sciences
Any 7-Chromatic Graphs Has K 7 Or K 4,4 As A Minor
Combinatorica
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
New approximation guarantee for chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
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Locally planar graphs are 5-choosable
Journal of Combinatorial Theory Series B
Approximating List-Coloring on a Fixed Surface
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
List-coloring graphs without subdivisions and without immersions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Structure theorem and isomorphism test for graphs with excluded topological subgraphs
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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It is known that computing the list chromatic number is harder than computing the chromatic number (assuming NP ≠ coNP). In fact, the problem of deciding whether a given graph is f-list-colorable for a function f: V → {c -1, c} for c ≥ 3 is Πp2-complete. In general, it is believed that approximating list coloring is hard for dense graphs. In this paper, we are interested in sparse graphs. More specifically, we deal with nontrivial minor-closed classes of graphs, i.e., graphs excluding some Kk minor. We refine the seminal structure theorem of Robertson and Seymour, and then give an additive approximation for list-coloring within k - 2 of the list chromatic number. This improves the previous multiplicative O(k)-approximation algorithm [20]. Clearly our result also yields an additive approximation algorithm for graph coloring in a minor-closed graph class. This result may give better graph colorings than the previous multiplicative 2-approximation algorithm for graph coloring in a minor-closed graph class [6]. Our structure theorem is of independent interest in the sense that it gives rise to a new insight on well-connected H-minor-free graphs. In particular, this class of graphs can be easily decomposed into two parts so that one part has bounded treewidth and the other part is a disjoint union of bounded-genus graphs. Moreover, we can control the number of edges between the two parts. The proof method itself tells us how knowledge of a local structure can be used to gain a global structure, which gives new insight on how to decompose a graph with the help of local-structure information.