On The Circumference Of 2-connected P₃-dominated Graphs

Let G be a connected graph. For x,y∈G with 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) = (?), G is P3-dominated(P3D) if it satisfies J(x,y)∪J (x,y) =(?), for every pair (x,y) of vertices at distance 2. Certainly P3D contains QCF as a subclass.In this paper, we prove that the circumference of a 2-connected P3-dominated graph G onn vertices is at least min{3δ+ 2,n} or G∈F∪{K2,3,K1,1,3}. Moreover if n≤4δthen Gis hamiltonian or G∈F∪{K2,3,K1,1,3}, where F is a class of 2-connected nonhamiltoniangraphs.