| A 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 τc(G), is the cardinality of a minimum clique-transversal set in G. A independent set I of a graph G is a subset of vertices of G such that no two vertices in the subset are joined by an edge in G. The independent number, denoted by α(G), is the cardinality of a maximum independent set of G. An open neighborhood set N(v) of v is the set of neighbors of v in G. Let H is a graph. G is a H-neighborhood graph if G[N(v)]～= H for any v ∈ V(G).Such as, if G is a 2K2-neighborhood graph, then G[N(v)]～= 2K2 for any v ∈ V(G). Wang et al.(Acta Math. Sin.(Engl. Ser.) 30(2014) 505-516) proved that τc(G) = ?n/3?for any 2-connected {K1,3, K4}-free 4-regular graphs of order n, and conjectured thatτc(G) ≤(10n + 3)/27 for a connected {K1,3, K4}-free 4-regular graph of order n. Kang et al.(2013) proved that τc(G) = ?n/3? for any 2-connected {K1,3, K4}-free 4-regular graphs of ordern. Furthermore, they also conjectured that α(G) ≥(8n- 3)/27 for a connected{K1,3, K4}-free 4-regular graphs of order n.In this paper, we show that the two conjectures are true, apart from only three exceptions, respectively. Besides, we prove that τc(G) ≤(13n + 3)/36 for any 2K2-neighborhood graph. The bound is better that(10n + 3)/27. Finally, we characterized the P4-neighborhood graph and the C4-neighborhood graph. |
