Graph minors. VI. Disjoint paths across a disc
Journal of Combinatorial Theory Series B
Graph minors. VII. Disjoint paths on a surface
Journal of Combinatorial Theory Series B
Gro¨tzsch's 3-color theorem and its counterparts for the torus and the projective plane
Journal of Combinatorial Theory Series B
Discrete Applied Mathematics - Special issue: Combinatorial Optimization 1992 (CO92)
Three-coloring graphs embedded on surfaces with all faces even-sided
Journal of Combinatorial Theory Series B
Journal of Graph Theory
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
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Coloring locally bipartite graphs on surfaces
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
Short path queries in planar graphs in constant time
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Graph Theory With Applications
Graph Theory With Applications
Linear time low tree-width partitions and algorithmic consequences
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Finding Shortest Non-Separating and Non-Contractible Cycles for Topologically Embedded Graphs
Discrete & Computational Geometry
Grad and classes with bounded expansion I. Decompositions
European Journal of Combinatorics
Grad and classes with bounded expansion II. Algorithmic aspects
European Journal of Combinatorics
Algorithms for finding an induced cycle in planar graphs and bounded genus graphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Three-coloring triangle-free planar graphs in linear time
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
List-color-critical graphs on a fixed surface
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
A shorter proof of the graph minor algorithm: the unique linkage theorem
Proceedings of the forty-second ACM symposium on Theory of computing
Three-coloring triangle-free planar graphs in linear time
ACM Transactions on Algorithms (TALG)
The disjoint paths problem in quadratic time
Journal of Combinatorial Theory Series B
Recognizing 3-colorings cycle-patterns on graphs
Pattern Recognition Letters
Testing first-order properties for subclasses of sparse graphs
Journal of the ACM (JACM)
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Gimbel and Thomassen asked whether 3-colorability of a triangle-free graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a linear-time algorithm for every surface which combined with previous results gives a lineartime algorithm to compute the chromatic number of such graphs. Our algorithm is based on a structure theorem that for a triangle-free graph drawn on a surface Σ guarantees the existence of a subgraph H, whose size depends only on Σ, such that there is an easy test whether a 3-coloring of H extends to a 3-coloring of G. The test is based on a topological obstruction, called the "winding number" of a 3-coloring. To prove the structure theorem we make use of disjoint paths with specified ends to find a 3-coloring. If the input triangle-free graph G drawn in Σ is 3-colorable we can find a 3-coloring in quadratic time, and if G quadrangulates Σ then we can find the 3-coloring in linear time. The latter algorithm requires two ingredients that may be of independent interest: a generalization of a data structure of Kowalik and Kurowski to weighted graphs and a speedup of a disjoint paths algorithm of Robertson and Seymour to linear time.