Hamiltonicity in 2-connected graphs with claws Original Research Article

Matthews and Sumner have proved in [12] that if G is a 2-connected claw-free graph of order n such that δ ⩾ (n − 2)/3, then G is hamiltonian. Li has shown that the bound for the minimum degree δ can be reduced to n/4 under the additional condition that G is not in Π, where Π is a class of graphs defined in [7]. On the other hand, we say that a graph G is almost claw-free if the centres of induced claws are independent and their neighbourhoods are 2-dominated. Broersma, Ryjáček and Schiermeyer have proved that if G is 2-connected almost claw-free graph of order n such that δ ⩾ (n − 2)/3, then G is hamiltonian. We generalize these results by considering the graphs whose claw centres are independent. If G is a 2-connected graph of order n and minimum degree δ such that n ⩽ 4δ − 3 and if the set of claw centres of G is independent, then we show that either G is hamiltonian or G ϵ F, where F is a class of graphs defined in the paper. The bound n ⩽ 4δ − 3 is sharp.