Pancyclicity in the Cartesian Product $(K_9-C_9 )^n$

TL;DR

This paper proves that the Cartesian product of a specific nearly complete graph with a cycle removed, taken multiple times, contains cycles of all lengths, demonstrating its pancyclicity.

Contribution

It establishes the pancyclicity of the Cartesian product of $(K_9 - C_9)$ taken n times, a novel result in graph theory.

Findings

$(K_9 - C_9)^n$ is pancyclic for all $n$

The result extends pancyclicity properties to complex graph products

Provides new insights into cycle structures in Cartesian products

Abstract

A graph $G$ on $m$ vertices is pancyclic if it contains cycles of length $l$, $3\leq l \leq m$ as subgraphs in $G$. The complete graph $K_{9}$ on 9 vertices with a cycle $C_{9}$ of length 9 deleted from $K_{9}$ is denoted by $(K_{9}-C_{9})$. In this paper, we prove that $(K_{9}-C_{9})^{n}$, the Cartesian product of $(K_{9}-C_{9})$ taken $n$ times, is pancyclic.