Number of colors needed to break symmetries of a graph by an arbitrary edge coloring

TL;DR

This paper introduces a new edge-coloring symmetry-breaking index for graphs, explores its properties for various graph families, and computes it for Cartesian products of prime graphs, extending understanding of graph automorphisms.

Contribution

It generalizes the concept of symmetry-breaking thresholds to edge-colorings, characterizes graphs with specific thresholds, and computes the index for Cartesian products of prime graphs.

Findings

=2 iff Ge1a1a1K_{1,2}

=3 iff Ge1a1a1P_4, K_{1,3} or K_3

Graphs with threshold 3 are infinite but not line graphs

Abstract

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph $G$, denoted by $\theta(G)$, is the minimum number of colors $k$ so that every $k$-coloring of $G$ is distinguishing. We generalize this concept to edge-coloring by defining an alternative index $\theta'(G)$. We consider $\theta'$ for some families of graphs and find its connection with edge-cycles of the automorphism group. Then we show that $\theta'(G)=2$ if and only if $G\simeq K_{1,2}$ and $\theta'(G)=3$ if and only if $G\simeq P_4, K_{1,3}$ or $K_3$. Moreover, we prove some auxiliary results for graphs whose distinguishing threshold is 3 and show that although there are infinitely many such graphs, but they are not line graphs. Finally, we compute $\theta'(G)$ when $G$ is the Cartesian product of simple prime graphs.