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

A clique of a graph GG is a complete subgraph maximal under inclusion and having at least two vertices. A clique-transversal set DD of a graph GG is a set of vertices of GG such that DD meets all cliques of GG. The clique-transversal number, denoted by τc(G)τc(G), is the cardinality of a minimum clique-transversal set in GG. Wang et al. (2014) proved that τc(G)=⌈n3⌉ for any 2-connected {K1,3,K4}{K1,3,K4}-free 4-regular graph of order nn, and conjectured that τc(G)≤10n+327 for a connected {K1,3,K4}{K1,3,K4}-free 4-regular graph of order nn.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.