Non-Hamiltonian $\frac{3}{2}$-Tough Plane Triangulations

Abstract

By Tutte's classic theorem of 1956 that every 4-connected planar graph is Hamiltonian, every planar graph of order at least three with toughness greater than $\frac{3}{2}$ is Hamiltonian. In 1999, Owens constructed a sequence of maximal planar graphs whose toughness approaches $\frac{3}{2}$ from below and which do not contain even a 2-factor, and he asked whether there exists a maximal planar graph with toughness exactly $\frac{3}{2}$ and with no 2-factor. In 2025, Shan constructed a $\frac{3}{2}$-tough plane triangulation with no 2-factor. In that construction, there are many pairs of vertices of degree $3$ that have a common neighbor. By imposing a distance condition on the vertices of degree $3$, Hao, Ma, Shan, and Yang recently proved that every $\frac{3}{2}$-tough plane triangulation of order at least three whose vertices of degree $3$ are pairwise at distance at least $3$ has a 2-factor, and they asked whether every such graph is Hamiltonian. We answer this question in the negative, and in fact prove the following stronger statement: for every positive integer $\ell$, there exists a $\frac{3}{2}$-tough non-Hamiltonian plane triangulation whose vertices of degree $3$ are pairwise at distance at least $\ell$. Thus, although the distance condition guarantees the existence of a 2-factor, it does not guarantee that the graph is Hamiltonian: the essential obstruction to a Hamiltonian cycle is a certain local configuration involving a vertex of degree $3$, rather than the proximity of such configurations in the graph.