Rainbow connection number and independence number of a graph

TL;DR

This paper explores the relationship between the rainbow connection number and the independence number of a connected graph, establishing an upper bound and demonstrating its tightness with examples.

Contribution

It proves that for any connected graph, the rainbow connection number is at most twice the independence number minus one, and shows this bound is tight.

Findings

Established the bound rc(G) ≤ 2α(G) - 1 for connected graphs.

Provided examples where this bound equals the diameter of the graph.

Showed the bound is tight by matching it with the diameter.

Abstract

Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum number of colors that are needed to make $G$ rainbow connected is called the rainbow connection number of $G$, denoted by $rc(G)$. In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if $G$ is a connected graph, then $rc(G)\leq 2\alpha(G)-1$. Two examples $G$ are given to show that the upper bound $2\alpha(G)-1$ is equal to the diameter of $G$, and therefore the best possible since the diameter is a lower bound of $rc(G)$.