Clique-transversal number in cubic graphs

A clique-transversal set S of a graph G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted τC(G), is the minimum cardinality of a clique-transversal set in G. In this paper we present an upper bound and a lower bound on τC(G) for a connected cubic graph G and characterize the extremal cubic graphs achieving the lower bound. Also, we show that every connected claw-free cubic graph G is weakly 2-colorable, this implies that it has clique-transversal number at most one-half its order.