Sharp upper bound for the rainbow connection numbers of 2-connected graphs

TL;DR

This paper establishes a sharp upper bound of loor(n/2)or the rainbow connection number of any 2-connected graph with n vertices, improving previous bounds to the best possible.

Contribution

It provides the first sharp upper bound for the rainbow connection number specifically for 2-connected graphs, refining earlier results.

Findings

The bound loor(n/2)or rc(G) is tight.

The result applies to all 2-connected graphs of order n.

This improves previous bounds by Caro et al.

Abstract

An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph $G$ is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper, we give a sharp upper bound that $rc(G)\leq\lceil\frac{n}{2}\rceil$ for any 2-connected graph $G$ of order $n$, which improves the results of Caro et al. to best possible.