The Perfect Matching Hamiltonian property in Prism and Crossed Prism graphs

TL;DR

This paper investigates the Perfect Matching Hamiltonian property in Prism and Crossed Prism graphs, showing that most Prism graphs lack this property except for the Cube, and identifying conditions under which Crossed Prism graphs possess it.

Contribution

The paper establishes that Prism graphs are generally not PMH except for the Cube, and determines for which parameters Crossed Prism graphs are PMH, advancing understanding of Hamiltonian properties in these graph classes.

Findings

Prism graphs are not PMH except for the Cube.

Conditions are identified when Crossed Prism graphs are PMH.

Provides insight into Hamiltonian cycles in specific graph families.

Abstract

A graph $G$ has the \emph{Perfect Matching Hamiltonian property} (or for short, $G$ is $PMH$) if, for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. In this note, we show that \emph{Prism graphs} $\cP_n$ are not $PMH$, except for the $Cube\ graph$, and indicate for which values of $n$ the \emph{Crossed Prism graphs} $\cCP_n$ are $PMH$.