Rainbow Connection Number and Connected Dominating Sets

TL;DR

This paper establishes new upper bounds for the rainbow connection number of connected graphs based on parameters like connected domination number and minimum degree, improving previous bounds and solving open problems.

Contribution

It introduces novel bounds relating rainbow connection number to connected domination number and minimum degree, extending results to various graph classes and solving an open problem.

Findings

Rainbow connection number is bounded by connected domination number plus 2 for graphs with minimum degree at least 2.

Bounds for specific graph classes like interval and chordal graphs are derived and shown to be tight.

A new upper bound of 3n/({} + 1) + 3 for the rainbow connection number is established, improving previous results.

Abstract

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, 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 show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).