Stability Number and k-Hamiltonian [a, b]-factors

Abstract

Let a, b and k be three integers with $b>a\ge 2$ and $k\ge 0$, and let G be a graph. If $G-U$ contains a Hamiltonian cycle for any $U\subseteq V(G)$ with $|U|=k$, then G is called a k-Hamiltonian graph. An [a, b]-factor F of a graph G is Hamiltonian if F admits a Hamiltonian cycle. If $G-U$ includes a Hamiltonian [a, b]-factor for every subset $U\subseteq V(G)$ with $|U|=k$, then we say that G has a k-Hamiltonian [a, b]-factor. In this paper, we prove that if G is a k-Hamiltonian graph with

$$ 1\le \alpha (G)\le \frac{4(b-2)(\delta (G)-a-k+1)}{(a+k+1)^{2}}, $$

then G admits a k-Hamiltonian [a, b]-factor. Furthermore, it is shown that this result is sharp.

Supported by the National Natural Science Foundation of China (Grant Nos. 11371009, 11501256, 61503160) and the National Social Science Foundation of China (Grant No. 14AGL001) and the Natural Science Foundation of Xinjiang Province of China (Grant No. 2015211A003), and sponsored by 333 Project of Jiangsu Province.