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

TL;DR

This paper determines the minimum toughness required for R-free graphs, where R is a linear forest of 5-7 vertices, to guarantee the existence of a 2-factor, extending known results for specific forbidden subgraphs.

Contribution

It identifies sharp toughness bounds for R-free graphs with R as a linear forest of 5 to 7 vertices to ensure 2-factors, generalizing previous results for specific cases.

Findings

Established sharp toughness bounds for various linear forests R.

Extended known results from specific forbidden subgraphs to broader classes.

Provided conditions under which R-free graphs contain 2-factors.

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 $3/2$-tough 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.