Toughness and existence of $2$-factors

Abstract

A graph is $t$-tough if the deletion of any set of, say, $m$ vertices from the graph leaves a graph with at most $\frac{m}{t}$ components. In 1973, Chv\'{a}tal suggested the problem of relating toughness to factors in graphs. In 1985, Enomoto et al. showed that each $2$-tough graph with at least three vertices has a $2$-factor, but for any $\epsilon>0$, there exists a $(2-\epsilon)$-tough graph on at least $3$ vertices having no $2$-factor. In recent years, the study of sufficient conditions for graphs with toughness less than $2$ having a $2$-factor has received a paramount interest. In this paper, we give new tight sufficient conditions for a $t$-tough graph having a $2$-factor when $1\le t<2$ by involving independence number, minimum degree, connectivity and forbidden forests.