2-factors in $\frac{3}{2}$-tough maximal planar graphs

TL;DR

This paper investigates conditions under which 3-vertices in 1.5-tough maximal planar graphs guarantee the existence of a 2-factor, extending classical results on graph toughness and factors.

Contribution

It provides a new sufficient condition involving the distance between degree-3 vertices for the existence of 2-factors in 1.5-tough maximal planar graphs.

Findings

Establishes a bound on the distance between degree-3 vertices ensuring 2-factors.

Extends classical results relating toughness and 2-factors in planar graphs.

Answers an open question about the existence of 2-factors in 1.5-tough maximal planar graphs.

Abstract

The toughness of a graph $G$ is defined as the minimum value of $|S|/c(G-S)$ over all cutsets $S$ of $G$ if $G$ is noncomplete, and is defined to be $\infty$ if $G$ is complete. For a real number $t$, we say that $G$ is $t$-tough if its toughness is at least $t$. Followed from the classic 1956 result of Tutte, every more than $\frac{3}{2}$-tough planar graph on at least three vertices has a 2-factor. In 1999, Owens constructed a sequence of maximal planar graphs with toughness $\frac{3}{2}-\varepsilon$ for any $\varepsilon >0$, but the graphs do not contain any 2-factor. He then posed the question of whether there exists a maximal planar graph with toughness exactly $\frac{3}{2}$ and with no 2-factor. This question was recently answered affirmatively by the third author. This naturally leads to the question: under what conditions does a $\frac{3}{2}$-tough maximal planar graph contain a 2-factor? In this paper, we provide a sufficient condition for the existence of 2-factors in $\frac{3}{2}$-tough maximal planar graphs, stated as a bound on the distance between vertices of degree 3.