Rainbow connectivity of randomly perturbed graphs

TL;DR

This paper proves that adding a large enough number of edges and coloring them with at least five colors in a perturbed graph ensures rainbow connectivity between all vertex pairs with high probability, confirming a conjecture.

Contribution

It establishes that for sufficiently many added edges and five or more colors, the resulting graph is rainbow connected with high probability, extending previous results.

Findings

Rainbow connectivity holds with high probability for large enough m and r ≥ 5.

Confirms a conjecture by Anastos and Frieze regarding rainbow connectivity.

Extends the understanding of rainbow connectivity in randomly perturbed graphs.

Abstract

In this note we examine the following random graph model: for an arbitrary graph $H$, with quadratic many edges, construct a graph $G$ by randomly adding $m$ edges to $H$ and randomly coloring the edges of $G$ with $r$ colors. We show that for $m$ a large enough constant and $r \geq 5$, every pair of vertices in $G$ are joined by a rainbow path, i.e., $G$ is {\it rainbow connected}, with high probability. This confirms a conjecture of Anastos and Frieze [{\it J. Graph Theory} {\bf 92} (2019)] who proved the statement for $r \geq 7$ and resolved the case when $r \leq 4$ and $m$ is a function of $n$.