Characterization and enumeration of 3-regular permutation graphs

TL;DR

This paper characterizes and enumerates all 3-regular permutation graphs, demonstrating their infinite variety and providing a construction method for all such graphs on any number of vertices.

Contribution

It introduces a construction method that characterizes all 3-regular permutation graphs and proves their infinite existence.

Findings

All 3-regular permutation graphs can be constructed using a specific method.

There are infinitely many connected 3-regular permutation graphs for any degree r ≥ 3.

Complete enumeration of 3-regular permutation graphs on n vertices is provided.

Abstract

A permutation graph is a graph that can be derived from a permutation, where the vertices correspond to letters of the permutation, and the edges represent inversions. We provide a construction to show that there are infinitely many connected $r$-regular permutation graphs for $r \geq 3$. We prove that all $3$-regular permutation graphs arise from a similar construction. Finally, we enumerate all $3$-regular permutation graphs on $n$ vertices.