Forbidden subgraphs and 2-factors in 3/2-tough graphs

TL;DR

This paper proves that certain tough and forbidden subgraph conditions guarantee the existence of a 2-factor in graphs, establishing sharp bounds and demonstrating the optimality of the forbidden subgraph.

Contribution

It establishes a new sufficient condition for 2-factors in 3/2-tough graphs with specific forbidden subgraphs, and shows the bounds are tight.

Findings

Every 3/2-tough (P4 ∪ P10)-free graph has a 2-factor.

The toughness condition is sharp for this property.

There exist (2-ε)-tough 2P5-free graphs without a 2-factor.

Abstract

A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ is a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free graph has a 2-factor. The toughness condition of this result is sharp. Moreover, for any $\varepsilon>0$ there exists a $(2-\varepsilon)$-tough $2P_5$-free graph without a 2-factor. This implies that the graph $P_4 \cup P_{10}$ is best possible for a forbidden subgraph in a sense.