The distinguishing index of graphs with infinite minimum degree

TL;DR

This paper proves bounds on the distinguishing index for graphs with infinite minimum degree, showing that such graphs can be edge-colored with at most 2 or 3 colors to break all non-trivial automorphisms.

Contribution

It establishes new bounds on the distinguishing index for infinite degree graphs, confirming a conjecture for regular graphs and extending results to broader classes.

Findings

Connected infinite regular graphs have a distinguishing index at most 2.

Graphs with infinite minimum degree and limited high-degree vertices have a distinguishing index at most 3.

The results apply to graphs with infinite order up to the continuum.

Abstract

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $\alpha$-regular graph for some infinite cardinal $\alpha$ then $D'(G) \le 2$, proving a conjecture of Lehner, Pil\'{s}niak, and Stawiski. We also show that if $G$ is a graph with infinite minimum degree and at most $2^\alpha$ vertices of degree $\alpha$ for every infinite cardinal $\alpha$, then $D'(G) \le 3$. In particular, $D'(G) \le 3$ if $G$ has infinite minimum degree and order at most $2^{\aleph_0}$.