On color-critical ($P_{5},\overline{P}_5$)-free graphs

Harjinder S. Dhaliwal; Angèle M. Hamel; Chính T. Hoàng; Frédéric Maffray; Tyler J. D. McConnell; Stefan A. Panait

On color-critical ($P_{5},\overline{P}_5$)-free graphs

Combinatorics 2014-12-01 v2

Authors: Harjinder S. Dhaliwal , Angèle M. Hamel , Chính T. Hoàng , Frédéric Maffray , Tyler J. D. McConnell , Stefan A. Panait

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Abstract

A graph is $k$ -critical if it is $k$ -chromatic but each of its proper induced subgraphs is ( $k-1$ )-colorable. It is known that the number of $4$ -critical $P_5$ -free graphs is finite, but there is an infinite number of $k$ -critical $P_5$ -free graphs for each $k \geq 5$ . We show that the number of $k$ -critical $(P_5, \overline{P}_5)$ -free graphs is finite for every fixed $k$ . Our result implies the existence of a certifying algorithm for $k$ -coloring $(P_5, \overline{P}_5)$ -free graphs.

Keywords

color-critical graphs $P_5$-free graphs $\overline{P}_5$-free graphs $k$-coloring induced subgraphs certifying algorithm graph theory

Cite

@article{arxiv.1403.8027,
  title  = {On color-critical ($P_{5},\overline{P}_5$)-free graphs},
  author = {Harjinder S. Dhaliwal and Angèle M. Hamel and Chính T. Hoàng and Frédéric Maffray and Tyler J. D. McConnell and Stefan A. Panait},
  journal= {arXiv preprint arXiv:1403.8027},
  year   = {2014}
}

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13 pages, minor revisions made

On color-critical ($P_{5},\overline{P}_5$)-free graphs

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