Non-Hamiltonian 3/2-Tough Plane Triangulations

@misc{shan2026nonhamiltonian,
  title = {{Non-Hamiltonian $\frac32$-Tough Plane Triangulations}},
  author = {Songling Shan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05414},
  eprint = {2609.05414},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05414},
}

TY  - UNPB
TI  - Non-Hamiltonian 3/2-Tough Plane Triangulations
AU  - Songling Shan
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05414
UR  - https://arxiv.org/abs/2609.05414
AB  - 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 3/2 is Hamiltonian. In 1999, Owens constructed a sequence of maximal planar graphs whose toughness approaches 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 3/2 and with no 2-factor. In 2025, Shan constructed a 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 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 ℓ, there exists a 3/2-tough non-Hamiltonian plane triangulation whose vertices of degree $3$ are pairwise at distance at least ℓ. 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.
ER  -