A construction of a $\frac{3}{2}$-tough plane triangulation with no 2-factor

TL;DR

This paper constructs a maximal 1.5-tough planar graph that lacks a 2-factor, resolving a long-standing open question about the existence of such graphs at this toughness level.

Contribution

It provides the first explicit example of a maximal 1.5-tough planar graph without a 2-factor, filling a gap in the understanding of toughness and 2-factors in planar graphs.

Findings

Constructed a maximal 1.5-tough planar graph with no 2-factor.

Answered an open question about the existence of such graphs at toughness exactly 1.5.

Extended the understanding of the relationship between toughness and 2-factors in planar graphs.

Abstract

In 1956, Tutte proved the celebrated theorem that every 4-connected planar graph is hamiltonian. This result implies that every more than $\frac{3}{2}$-tough planar graph on at least three vertices is hamiltonian and so has a 2-factor. Owens in 1999 constructed non-hamiltonian maximal planar graphs of toughness arbitrarily close to $\frac{3}{2}$ and asked whether there exists a maximal non-hamiltonian planar graph of toughness exactly $\frac{3}{2}$. In fact, the graphs Owens constructed do not even contain a 2-factor. Thus the toughness of exactly $\frac{3}{2}$ is the only case left in asking the existence of 2-factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by constructing a maximal $\frac{3}{2}$-tough plane graph with no 2-factor, answering the question asked by Owens as well as by Bauer, Broersma, and Schmeichel.