Colorful monochromatic connectivity of random graphs

TL;DR

This paper investigates the threshold functions for the maximum number of colors in monochromatic connection colorings of Erdős-Rényi random graphs, establishing precise probabilistic thresholds based on the function f(n).

Contribution

It determines sharp threshold functions for the property that the monochromatic connection number exceeds a given function f(n) in random graphs.

Findings

Sharp threshold at p=(f(n)+n log log n)/n^2 for f(n) ≥ ℓ n log n.

Threshold at p=log n / n for f(n)=o(n log n).

Provides probabilistic bounds for monochromatic connectivity in Erdős-Rényi graphs.

Abstract

An edge-coloring of a connected graph $G$ is called a {\it monochromatic connection coloring} (MC-coloring, for short), introduced by Caro and Yuster, if there is a monochromatic path joining any two vertices of the graph $G$. Let $mc(G)$ denote the maximum number of colors used in an MC-coloring of a graph $G$. Note that an MC-coloring does not exist if $G$ is not connected, and in this case we simply let $mc(G)=0$. We use $G(n,p)$ to denote the Erd\"{o}s-R\'{e}nyi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independently from other pairs. For any function $f(n)$ satisfying $1\leq f(n)<\frac{1}{2}n(n-1)$, we show that if $\ell n \log n\leq f(n)<\frac{1}{2}n(n-1)$ where $\ell\in \mathbb{R}^+$, then $p=\frac{f(n)+n\log\log n}{n^2}$ is a sharp threshold function for the property $mc\left(G\left(n,p\right)\right)\ge f(n)$; if $f(n)=o(n\log n)$, then $p=\frac{\log n}{n}$ is a sharp threshold function for the property $mc\left(G\left(n,p\right)\right)\ge f(n)$.