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Erschienen in:
2016 | OriginalPaper | Buchkapitel
Exhaustive Generation of k-Critical \({\mathcal H}\)-Free Graphs
verfasst von : Jan Goedgebeur, Oliver Schaudt
Erschienen in: Graph-Theoretic Concepts in Computer Science
Verlag: Springer Berlin Heidelberg
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Abstract
We describe an algorithm for generating all k-critical \({\mathcal H}\)-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many 4-critical \((P_7,C_k)\)-free graphs, for both \(k=4\) and \(k=5\). We also show that there are only finitely many 4-critical \((P_8,C_4)\)-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the 3-colorability problem in the respective classes.
Moreover, we prove that for every t, the class of 4-critical planar \(P_t\)-free graphs is finite. We also determine all 52 4-critical planar \(P_7\)-free graphs. We also prove that every \(P_{11}\)-free graph of girth at least five is 3-colorable, and show that this is best possible by determining the smallest 4-chromatic \(P_{12}\)-free graph of girth at least five. Moreover, we show that every \(P_{14}\)-free graph of girth at least six and every \(P_{17}\)-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.