Kao's sharpness conjecture for Hamiltonicity of graph–path products

A path factor of a graph is a factor whose components are paths on at least two vertices. Let $P_m$ be the path on $m$ vertices, and let $\Delta(G)$ denote the maximum degree of a graph $G$. Kao's conjecture. For every integer $k\geq 3$, there exists a connected graph $G$ with a path factor and $\Delta(G)=k$ such that

$$P_{4k-4}\Box G$$

is not hamiltonian. This is the main result stated in the paper and is a more specific formulation of the sharpness phenomenon for the bound $n\geq 4\Delta(T)-2$; the paper presents it as a conjecture from Kao and proves it.