Proper connection number of random graphs

TL;DR

This paper proves that in the Erdős-Rényi random graph model, with high probability, the proper connection number is 2 for sufficiently large graphs when the edge probability exceeds a certain threshold.

Contribution

It establishes that almost all large random graphs have a proper connection number of 2 under specified probabilistic conditions.

Findings

Proper connection number is 2 for large graphs with p ≥ (log n + α(n))/n.

Almost all graphs have proper connection number 2.

Threshold for proper connection number in Erdős-Rényi graphs is identified.

Abstract

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one proper path in $G$. In this paper, we show that almost all graphs have the proper connection number 2. More precisely, let $G(n,p)$ 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 $p$ independent from other pairs. We prove that for sufficiently large $n$, $pc(G(n,p))\le2$ if $p\ge\frac{\log n +\alpha(n)}{n}$, where $\alpha(n)\rightarrow \infty$.