On the odd girth and the circular chromatic number of generalized Petersen graphs

TL;DR

This paper investigates the odd girth and circular chromatic number of generalized Petersen graphs, demonstrating that their odd girth can be computed via integer programming and establishing their odd-pentagonality.

Contribution

It introduces a method to compute the odd girth of generalized Petersen graphs using integer programming and proves the class is odd-girth-closed and odd-pentagonal.

Findings

Odd girth of generalized Petersen graphs can be explicitly computed.

The class of generalized Petersen graphs is odd-girth-closed.

The class of generalized Petersen graphs is shown to be odd-pentagonal.

Abstract

A class of simple graphs such as ${\cal G}$ is said to be {\it odd-girth-closed} if for any positive integer $g$ there exists a graph $G \in {\cal G}$ such that the odd-girth of $G$ is greater than or equal to $g$. An odd-girth-closed class of graphs ${\cal G}$ is said to be {\it odd-pentagonal} if there exists a positive integer $g^*$ depending on ${\cal G}$ such that any graph $G \in {\cal G}$ whose odd-girth is greater than $g^*$ admits a homomorphism to the five cycle (i.e. is $C_{_{5}}$-colorable). In this article, we show that finding the odd girth of generalized Petersen graphs can be transformed to an integer programming problem, and using this we explicitly compute the odd girth of such graphs, showing that the class is odd-girth-closed. Also, motivated by showing that the class of generalized Petersen graphs is odd-pentagonal, we study the circular chromatic number of such graphs.