Matthews and Sumner have proved in [10] that if G is a 2-connectedclaw-free graph of order n such that δ(G) ≥ (n-2)/3, then G isHamiltonian. We say that a graph is almost claw-free if for every vertexv of G, 〈N(v)〉 is 2-dominated and the set A of centers of claws of G isan independent set. Broersma et al. [5] have proved that if G isa 2-connected almost claw-free graph of order n such that n such that δ(G) ≥ (n-2)/3, then G is Hamiltonian. We generalize these resultsby considering the graphs satisfying the following property: for everyvertex v ∈ A, there exist exactly two vertices x and y of V\A such that N(v) ⊆ N[x] ∪ N[y]. We extend some other knownresults on claw-free graphs to this new class of graphs.
