Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

Abstract : A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.
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Discrete Mathematics and Theoretical Computer Science, DMTCS, 2009, 11 (2), pp.15--24
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Gábor Bacsó, Zsolt Tuza. Clique-transversal sets and weak 2-colorings in graphs of small maximum degree. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2009, 11 (2), pp.15--24. 〈hal-00988213〉

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