Discrete Mathematics & Theoretical Computer Science, Vol 11, No 2 (2009)

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

Gabor Bacso, Zsolt Tuza


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 ⌊ 19n/30+2/15 ⌋. 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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