A note on non-isomorphic edge-color classes in random graphs

TL;DR

This paper studies the maximum number of edge colorings in Erdős-Rényi random graphs where no two color classes are isomorphic, exploring a novel graph coloring parameter in probabilistic graph theory.

Contribution

It introduces and analyzes the parameter (G) for Erd
51s-Rf3ny random graphs, providing new insights into non-isomorphic edge-color classes.

Findings

Characterizes the typical behavior of (G) in G(n,p)

Establishes bounds for (G) in different regimes of p

Connects (G) with graph symmetry and automorphisms

Abstract

For a graph $G$, let $\tau(G)$ be the maximum number of colors such that there exists an edge-coloring of $G$ with no two color classes being isomorphic. We investigate the behavior of $\tau(G)$ when $G=G(n, p)$ is the classical Erd\H{o}s-R\'enyi random graph.