Conflict-free connection number of random graphs

TL;DR

This paper proves that in the Erdős-Rényi random graph model, the conflict-free connection number is almost always 2 once the graph is sufficiently large and connected, highlighting a phase transition in graph coloring.

Contribution

It establishes that for large random graphs, the conflict-free connection number is at most 2 once the graph becomes connected with high probability.

Findings

Almost all large connected random graphs have cfc(G) = 2.

The threshold for the conflict-free connection number to be at most 2 is when p ≥ (log n + α(n))/n with α(n)→∞.

The result links connectivity phase transition to conflict-free coloring properties.

Abstract

An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. In this paper, we show that almost all graphs have the conflict-free connection number 2. More precisely, let $G(n,p)$ denote the Erd\H{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 $n$, $cfc(G(n,p))\le 2$ if $p\ge\frac{\log n +\alpha(n)}{n}$, where $\alpha(n)\rightarrow \infty$. This means that as soon as $G(n,p)$ becomes connected with high probability, $cfc(G(n,p))\le 2$.