A new approach to pancyclicity of Paley graphs I

TL;DR

This paper introduces a novel method for establishing the pancyclicity of Paley graphs by analyzing Cayley graphs and their Cartesian products, providing new proofs and insights into their cycle structures.

Contribution

It presents new theoretical results on the pancyclicity of Cayley graphs and Cartesian products, culminating in a new proof of Paley graph pancyclicity.

Findings

Cayley graphs can be pancyclic under certain conditions.

Cartesian products of specific graphs are pancyclic.

A new proof of Paley graph pancyclicity is provided.

Abstract

Let $G$ be an undirected graph of order $n$ and let $C_i$ be an $i$-cycle graph. $G$ is called pancyclic if $G$ contains a $C_i$ for any $i\in \{3,4,\ldots,n\}$. We show that the pancyclicity of specific Cayley graphs and the Cartesian product of specific two graphs. As a corollary of these two theorems, we provide a new proof of the pancyclicity of the Paley graph.