Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs

Abstract

For a given graph R, a graph G is R-free if G does not contain R as an induced subgraph. It is known that every 2-tough graph with at least three vertices has a 2-factor. In graphs with restricted structures, it was shown that every 2K 2 -free 3/2tough graph with at least three vertices has a 2-factor, and the toughness bound 3/2 is best possible. In viewing 2K 2 , the disjoint union of two edges, as a linear forest, in this paper, for any linear forest R on 5, 6, or 7 vertices, we find the sharp toughness bound t such that every t-tough R-free graph on at least three vertices has a 2-factor.