Odd extensions of transitive groups via symmetric graphs

TL;DR

This paper investigates when the presence of even automorphisms in symmetric graphs implies the existence of odd automorphisms, providing complete results for cubic symmetric graphs to understand their automorphism group structure.

Contribution

It offers a comprehensive analysis of odd automorphisms in symmetric graphs, especially cubic ones, advancing understanding of automorphism group extensions.

Findings

Complete characterization of odd automorphisms in cubic symmetric graphs

Conditions under which even automorphisms imply odd automorphisms

Enhanced understanding of automorphism group structures in symmetric graphs

Abstract

When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that existence of automorphisms acting as even permutations on the vertex set of a graph, called {\em even automorphisms}, forces existence of automorphisms that act as odd permutations, called {\em odd automorphisms}. As a first step towards resolving the above question, a complete information on existence of odd automorphisms in cubic symmetric graphs is given.