Rainbow connection number of dense graphs

TL;DR

This paper establishes tight bounds on the rainbow connection number for dense graphs based on their number of edges, showing it is at most 3 or 4 under specific density conditions.

Contribution

It provides new sharp bounds on the rainbow connection number for dense graphs depending on their edge count.

Findings

If |E(G)| ≥ (n-2 choose 2) + 2, then rc(G) ≤ 3.

If |E(G)| ≥ (n-3 choose 2) + 3, then rc(G) ≤ 4.

The bounds are proven to be sharp.

Abstract

An edge-colored graph $G$ is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we show that $rc(G)\leq 3$, if $|E(G)|\geq {{n-2}\choose 2}+2$, and $rc(G)\leq 4$, if $|E(G)|\geq {{n-3}\choose 2}+3$. These bounds are sharp.