Triangle-Free Graphs of Toughness Approaching Two Without a 2-Factor

Abstract

By work of Enomoto, Jackson, Katerinis, and Saito from 1985, every $2$-tough graph has a $2$-factor, and this toughness bound is best possible: for every $\varepsilon>0$, there exist $(2-\varepsilon)$-tough graphs with no $2$-factor. It is natural to ask whether the latter statement remains true for triangle-free graphs. Bauer, van den Heuvel, and Schmeichel conjectured this in 1996. In the same paper, they proposed an infinite family of triangle-free graphs with no $2$-factor whose toughness they believed approaches $2$, but the required toughness bound was not established. In this paper, we confirm their conjecture. For every even integer $q\ge 6$, we construct a triangle-free graph $G_q$ with no $2$-factor and with toughness \[ τ(G_q) =\frac{2q^2-q-2}{q^2+q} =2-\frac{3q+2}{q^2+q}. \] In particular, $τ(G_q)\to 2$ as $q\to\infty$, showing that the threshold $2$ for the existence of a $2$-factor remains best possible even within the class of triangle-free graphs.