Vizing's 2-factor Conjecture Involving Toughness and Maximum Degree Conditions

TL;DR

This paper investigates conditions involving toughness and maximum degree that guarantee the existence of a 2-factor in $ ext{Delta}$-critical graphs, advancing understanding of Vizing's conjecture.

Contribution

It establishes new sufficient conditions based on toughness and maximum degree for $ ext{Delta}$-critical graphs to contain a 2-factor, and introduces novel proof techniques.

Findings

If $G$ is $ ext{Delta}$-critical with toughness ≥ 3/2 and max degree ≥ n/3, then $G$ has a 2-factor.

Proves new sufficient conditions for 2-factors in $ ext{Delta}$-critical graphs.

Develops innovative methods for proving the existence of 2-factors.

Abstract

Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then $G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if $\Delta(G)\ge 2n/3+12$, then $G$ has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical graph under "moderate" given toughness and maximum degree conditions. In particular, we show that if $G$ is an $n$-vertex $\Delta$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. In addition, we develop new techniques in proving the existence of 2-factors in graphs.