The clique-transversal number of a{K1,3,K4}-free 4-regular graph

Abstract

a b s t r a c tA clique of a graph G is a complete subgraph maximal under inclusion and having at least two vertices. A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τ c (G), is the cardinality of a minimum clique-transversal set in G. proved that τ c (G) = ⌈ n 3 ⌉ for any 2-connected {K 1,3 , K 4 }-free 4-regular graph of order n, and conjectured that τ c (G) ≤ 10n+3 27 for a connected {K 1,3 , K 4 }-free 4-regular graph of order n.In this paper, we give a short proof of the aforementioned theorem of Wang et al. and show that the above conjecture is true, apart from only three exceptions.