On Rainbow Connection Number and Connectivity

TL;DR

This paper explores the relationship between the rainbow connection number of a graph and its vertex and edge connectivity, providing bounds, tightness results, and verifying conjectures for specific graph classes.

Contribution

It improves existing bounds on rainbow connection number in terms of vertex connectivity and confirms the conjecture for certain graph classes.

Findings

Bound in terms of edge connectivity is tight up to additive constants.

Bound in terms of vertex connectivity can be improved to a function involving epsilon.

Conjecture holds for graphs with vertex connectivity 2, chordal graphs, and graphs with girth at least 7.

Abstract

Rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colours needed to colour its edges, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum degree $\delta$, the rainbow connection number is upper bounded by $3n/(\delta + 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(\lambda + 1) + 3$ and $3n/(\kappa + 1) + 3$ for rainbow connection number where $\lambda$ and $\kappa$, respectively, denote the edge and vertex connectivity of the graph. We show that the above bound in terms of edge connectivity is tight up-to additive constants and show that the bound in terms of vertex connectivity can be improved to $(2 + \epsilon)n/\kappa + 23/ \epsilon^2$, for any $\epsilon > 0$. We conjecture that rainbow connection number is upper bounded by $n/\kappa + O(1)$ and show that it is true for $\kappa = 2$. We also show that the conjecture is true for chordal graphs and graphs of girth at least 7.