Neighborhood Union Conditions For Hamiltonicity Of P3-dominated Graphs

Let G be a connected graph. For x,y∈V(G) at distance 2, we define 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)}.G is quasi claw-free (QCF) if it satisfies J(x,y)≠φ, and G is P₃-dominated (P3D) if it satisfies J(x,y)∪J'(x,y)≠φfor every pair (x, y) of vertices at distance 2. Certainly, P3D contains QCF as a subclass, and hence contains all claw-free graphs. For a noncomplete graph G, the number NC and NC₂ are defined as NC = min{|N(x)∪N(y)| : x,y∈V(G) and xy (?) E(G)} and NC₂=min{|N(x)∪N(y)| : x,y∈V(G) and d(x,y)=2} , respectively. For a complete graph G, set NC = NC₂ = |V(G)| - 1. In this paper, we prove that a 2-connected P₃-dominated graph of order n is traceable if NC≥(n - 2)/2. Moreover, we prove that a 3-connected P₃-dominated graph of order n is hamiltonian if NC₂≥(2n- 6)/3. This extend the known results on claw-free graphs.