Neighborhood Union Conditions for Hamiltonicity of P3-dominated Graphs II

A graph G is called P-3-dominated (P3D) if it satisfies J(x, y) U J'(x, y) not equal empty set for every pair (x, y) of vertices at distance 2, where J(x, y) = {u/u is an element of N(x) boolean AND N(y), N[u] subset of N[x]UN[y]} and J'(x,y) = {u/u is an element of N(x)boolean AND N(y)| if v is an element of N(u)\(N[x]boolean OR N[y]), then (N(u)boolean OR N(x) boolean OR N (y))\{x, y,v} subset of N(v)} for x, y is an element of V(G) at distance 2}. For a noncomplete graph G, the number NC is defined as NC = min{|N(x) boolean OR N(y)| : x, y is an element of V(G) and xy is not an element of E(G)}, for a complete graph G, set NC = |V(G)|-1. In this paper, we prove that a 2-connected P3-dominated graph G of order n is hamiltonian if G is not an element of{K-2,K-3, K-1,K-1,K-3} and NC(G) >= (2n - 5)/3, moreover it is best possible.