Toughness for the Existence of \(k\)-Hamiltonian \([a, b]\)-Factors

Abstract

Let \(a\), \(b\), and \(k\) be three nonnegative integers with \(a \geq 2\) and \(b \geq a(k+1)+2\). A graph \(G\) is called a \(k\)-Hamiltonian graph if \(G – U\) contains a Hamiltonian cycle for every subset \(U \subseteq V(G)\) with \(|U| = k\). An \([a, b]\)-factor \(F\) of \(G\) is called a Hamiltonian \([a, b]\)-factor if \(F\) contains a Hamiltonian cycle. If \(G – U\) has a Hamiltonian \([a, b]\)-factor for every subset \(U \subseteq V(G)\) with \(|U| = k\), then we say that \(G\) admits a \(k\)-Hamiltonian \([a, b]\)-factor. Suppose that \(G\) is a \(k\)-Hamiltonian graph of order \(n\) with \(n \geq a+k+2\). In this paper, it is proved that \(G\) includes a \(k\)-Hamiltonian \([a, b]\)-factor if \(\delta(G) \geq a+k\) and \(t(G) \leq a-1+\frac{(a-1)(k+1)}{b-2}\).