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Exhaustive Generation of k-Critical \({\mathcal H}\)-Free Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9941))

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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.

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Acknowledgments

Several of the computations for this work were carried out using the Stevin Supercomputer Infrastructure at Ghent University. Jan Goedgebeur is supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO).

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Authors and Affiliations

  1. Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Ghent, Belgium

    Jan Goedgebeur

  2. Institut Für Informatik, Universität Zu Köln, Cologne, Germany

    Oliver Schaudt

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  1. Jan Goedgebeur
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  2. Oliver Schaudt
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Correspondence to Jan Goedgebeur .

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Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Goedgebeur, J., Schaudt, O. (2016). Exhaustive Generation of k-Critical \({\mathcal H}\)-Free Graphs. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_10

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  • DOI: https://doi.org/10.1007/978-3-662-53536-3_10

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