Toughness and \([a, b]\)-Factors with Prescribed Properties

Abstract

In this paper, we consider the relationship between toughness and the existence of \([a, b]\)-factors with inclusion/exclusion properties. We obtain that if \(t(G) \geq a – 1 + \frac{a – 1}{b}\) with \(b > a > 2\), where \(a, b\) are two integers, then for any two given edges \(e_1\) and \(e_2\), there exist an \([a, b]\)-factor including \(e_1, e_2\); and an \([a, b]\)-factor including \(e_1\) and excluding \(e_2\); as well as an \((a, b)\)-factor excluding \(e_1, e_2\). Furthermore, it is shown that the results are best possible in some sense.