Hamiltonicity in Partly claw-free graphs

Matthews and Sumner have proved in [10] that if G is a 2-connected claw-free graph of order n such that δ(G) ≥ (n − 2)/3, then G is Hamiltonian. We say that a graph is almost claw-free if for every vertex v of G, 〈N(v)〉 is 2-dominated and the set A of centers of claws of G is an independent set. Broersma et al. [5] have proved that if G is a 2-connected almost claw-free graph of order n such that δ(G) ≥ (n− 2)/3, then G is Hamiltonian. We generalize these results by considering the graphs satisfying the following property: for every vertex 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 known results on claw-free graphs to this new class of graphs.