Asymmetric edge-coloring of graphs with simple automorphism group

TL;DR

This paper proves that finite graphs with a simple automorphism group can be edge-colored with just two colors to break all non-trivial automorphisms, establishing a key property of such graphs.

Contribution

It establishes that graphs with simple automorphism groups have a distinguishing index of 2, providing a new insight into symmetry-breaking in graph automorphisms.

Findings

Graphs with simple automorphism groups have D'(b4) = 2

Edge-coloring with 2 colors suffices to break all automorphisms in these graphs

The result applies to finite graphs with simple automorphism groups

Abstract

The distinguishing index $D'(\Gamma)$ of a graph $\Gamma$ is the least number $k$ such that $\Gamma$ has an edge-coloring with $k$ colors preserved only by the trivial automorphism. In this paper we prove that if the automorphism group of a finite graph $\Gamma$ is simple, then its distinguishing index $D'(\Gamma)=2$.