Elusive groups of automorphisms of digraphs of small valency

TL;DR

This paper investigates elusive automorphism groups in small-valency graphs and digraphs, proving their non-existence in certain cases and classifying all such groups in others, advancing understanding of symmetry in low-valency structures.

Contribution

It proves that no elusive automorphism group exists for connected graphs of valency at most four and classifies all elusive automorphism groups for connected digraphs with out-valency at most three.

Findings

No elusive automorphism groups for connected graphs of valency ≤ 4.

Complete classification of elusive automorphism groups for certain digraphs.

Identification of structural properties preventing elusive automorphisms.

Abstract

A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.