Hamiltonicity of Cartesian products of graphs

TL;DR

This paper proves a conjecture regarding the Hamiltonicity of Cartesian products of graphs, specifically showing that for graphs with maximum degree  or more, certain products are not Hamiltonian under specified conditions.

Contribution

The paper confirms a conjecture that for graphs with maximum degree  or higher, there exist graphs with a path factor where their Cartesian product with certain paths is not Hamiltonian.

Findings

Proved the conjecture about non-Hamiltonian Cartesian products for graphs with degree .

Established conditions under which the Cartesian product of a graph and a path is not Hamiltonian.

Extended understanding of Hamiltonicity in graph products involving graphs with specific degree and path length constraints.

Abstract

A path factor in a graph $G$ is a factor of $G$ in which every component is a path on at least two 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, $n$ is an even integer and $n\geq 4\Delta (T)-2$. They conjectured that for every $\Delta \geq 3$ there exists a graph $G$ of maximum degree $\Delta$ which has a path factor, such that for every even $n< 4\Delta-2$ the product $G\Box P_n$ is not hamiltonian. In this article we prove this conjecture.