Graph minors. VII. Disjoint paths on a surface
Journal of Combinatorial Theory Series B
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Five-coloring maps on surfaces
Journal of Combinatorial Theory Series B
List colourings of planar graphs
Discrete Mathematics
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
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
You can't paint yourself into a corner
Journal of Combinatorial Theory Series B
Long cycles in graphs on a fixed surface
Journal of Combinatorial Theory Series B
A new lower bound on the number of edges in colour-critical graphs and hypergraphs
Journal of Combinatorial Theory Series B
The chromatic number of a graph of girth 5 on a fixed surface
Journal of Combinatorial Theory Series B
Exponentially many 5-list-colorings of planar graphs
Journal of Combinatorial Theory Series B
Graph and map isomorphism and all polyhedral embeddings in linear time
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Dirac's map-color theorem for choosability
Journal of Graph Theory
Approximating List-Coloring on a Fixed Surface
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Coloring triangle-free graphs on surfaces
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Hadwiger's conjecture is decidable
Proceedings of the forty-first annual ACM symposium on Theory of computing
List-coloring graphs without subdivisions and without immersions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Time optimal d-list colouring of a graph
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
Graphs with Two Crossings Are 5-Choosable
SIAM Journal on Discrete Mathematics
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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-list-assignment. We prove the following conjecture posed by Thomassen in 1994: "There are only finitely many list-color-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-list-assignment 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-list-assignment L, where k ≥ 5. Question: Does G admit an L-coloring? If not, provide a certificate for this. If yes, then return an L-coloring. 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-list-colorings for every 5-list-assignment for G." Thomassen himself proved that this conjecture holds for usual 5-colorings. 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.