On the Pythagorean Holes of Certain Graphs

TL;DR

This paper introduces the concept of Pythagorean holes in graphs, exploring their properties and analyzing their occurrence in set-graphs and Jaco graphs, expanding the understanding of primitive cycles related to Pythagorean triples.

Contribution

It defines Pythagorean holes in graphs and investigates their properties, including in specific graph classes like set-graphs and Jaco graphs, providing new insights into graph cycles.

Findings

Defined Pythagorean holes in graphs.

Analyzed properties of Pythagorean holes.

Studied their occurrence in set-graphs and Jaco graphs.

Abstract

A \emph{primitive hole} of a graph $G$ is a cycle of length 3 in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of the graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is the number of primitive holes incident on the vertex $v$. In this paper, we introduce the notion of Pythagorean holes of graphs and initiate some interesting results on Pythagorean holes in general as well as results in respect of set-graphs and Jaco graphs.