Hamiltonicity of Cartesian products of trees with odd paths

Irena Hrastnik Ladinek; Tjasa Paj Erker; Simon Spacapan

Hamiltonicity of Cartesian products of trees with odd paths

Combinatorics 2026-07-10 v1

Authors: Irena Hrastnik Ladinek , Tjasa Paj Erker , Simon Spacapan

Abstract

A $\{P_2,P_3\}$ -factor in a graph $G$ is a factor of $G$ in which every component is a path on two or three vertices. Let $T\Box P_n$ be the Cartesian product of a tree $T$ and a path on $n$ vertices. Kao and Weng proved that $T\Box P_n$ is hamiltonian if $T$ has a path factor and $n$ is a sufficiently large even integer. In this article we prove that, for every odd $n$ , there exists a tree $T$ of maximum degree 4 that has a $\{P_2,P_3\}$ -factor such that $T\Box P_n$ is not hamiltonian, thereby refuting a conjecture by Kao and Weng.

Keywords

Hamiltonian cycles Cartesian products path factors trees graph theory

Cite

@article{arxiv.2607.09270,
  title  = {Hamiltonicity of Cartesian products of trees with odd paths},
  author = {Irena Hrastnik Ladinek and Tjasa Paj Erker and Simon Spacapan},
  journal= {arXiv preprint arXiv:2607.09270},
  year   = {2026}
}