Degree Conditions for k-Hamiltonian [a, b]-factors

Abstract

Let a, b, k be nonnegative integers with 2 ≤ a < 6. A graph G is called a k-Hamiltonian graph if G − U contains a Hamiltonian cycle for any subset U ⊆ V(G) with ∣U∣ = k. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. If G − U admits a Hamiltonian [a, b]-factor for any subset U ⊆ V(G) with ∣U∣ = k, then we say that G has a k-Hamiltonian [a, b]-factor. Suppose that G is a k-Hamiltonian graph of order n with $n\geq\frac{(a+b-4)(2a+b+k-6)}{b-2}+k$ and δ(G) ≥ a + k. In this paper, it is proved that G admits a k-Hamiltonian [a, b]-factor if $\max\{d_{G}(x),d_{G}(y)\}\geq\frac{(a-2)n+(b-2)k}{a+b-4}+2$ for each pair of nonadjacent vertices x and y in G.