Proper Distinguishing Colorings with Few Colors for Graphs with Girth at Least 5

TL;DR

This paper proves a conjecture that connected graphs with girth at least 5 (excluding C6) can be properly distinguished with at most Δ+1 colors, where Δ is the maximum degree.

Contribution

The paper confirms Collins and Trenk's conjecture, establishing an upper bound on the distinguishing chromatic number for a broad class of graphs.

Findings

Proved the conjecture for graphs with girth at least 5.

Established that the bound is tight for certain graphs.

Extended understanding of symmetry-breaking colorings in graphs.

Abstract

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map. Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $\chi_D(G)\le \Delta+1$. We prove this conjecture.