Degree conditions and cycle extendability Original Research Article

A non-Hamiltonian cycle C in a graph G is extendable if there is a cycle C′ in G with V(C′) ⊃ V(C) with one more vertex than C. For any integer k ⩾ 0, a cycle C is k-chord extendable if it is extendable to the cycle C′ using at most k of the chords of the cycle C. It will be shown that if G is a graph of order n, then δ(G) > 3n/4 − 1 implies that any proper cycle is 0-chord extendable, δ(G) > 5n/9 implies that any proper cycle is 1-chord extendable, and δ(G) > [n/2] implies that any proper cycle is 2-chord extendable. Also, each of these results is sharp in the sense that the minimum degree condition cannot, in general, be lowered.