Neighborhood Union Conditions for Hamiltonicity of P 3-Dominated Graphs

A graph G is called P3-dominated (P3D) if it satisfies J(x, y) ∪ J ′(x, y) 6= ∅ for every pair (x, y) of vertices at distance 2, where J(x, y) = {u|u ∈ N(x) ∩N(y), N [u] ⊆ N [x]∪N [y]} and J ′(x, y) = {u|u ∈ N(x)∩N(y)| if v ∈ N(u)\(N [x]∪N [y]), then (N(u)∪ N(x) ∪ N(y))\{x, y, v} ⊆ N(v)} for x, y ∈ V (G) at distance 2}. For a noncomplete graph G, the number NC is defined as NC = min{|N(x) ∪ N(y)| : x, y ∈ V (G) and xy / ∈ 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 / ∈ {K2,3,K1,1,3} and NC(G) ≥ (2n− 5)/3, moreover it is best possible.