Rainbow connection number and the number of blocks

TL;DR

This paper establishes tight upper bounds on the rainbow connection number for connected graphs with cut vertices and for bridgeless graphs, linking these bounds to the graph's block structure and connectivity.

Contribution

It introduces new tight upper bounds for the rainbow connection number based on the number of blocks with even order and connectivity properties.

Findings

For graphs with cut vertices, $rc(G) \,\leq\, (n + r - 1)/2$, where $r$ is the number of even-ordered blocks.

For 2-edge-connected (bridgeless) graphs, the paper provides a tight upper bound on $rc(G)$.

The bounds are proven to be tight, meaning they are the best possible.

Abstract

An edge-colored graph $G$ is rainbow connected if every pair of vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of $G$ is defined to be the minimum integer $t$ such that there exists an edge-coloring of $G$ with $t$ colors that makes $G$ rainbow connected. For a graph $G$ without any cut vertex, i.e., a 2-connected graph, of order $n$, it was proved that $rc(G)\leq \lceil\frac n 2 \rceil$ and the bound is tight. In this paper, we prove that for a connected graph $G$ of order $n$ with cut vertices, $rc(G) \leq\frac{n+r-1} 2$, where $r$ is the number of blocks of $G$ with even orders, and the upper bound is tight. Moreover, we also obtain a tight upper bound for a bridgeless graph, i.e., a 2-edge-connected graph.