On Rainbow Connection Number and Connectivity
L.Sunil Chandran, Rogers Mathew, and Deepak Rajendraprasad

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, , of a connected graph 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 , the rainbow connection number is upper bounded by [Chandran et al., 2010]. This directly gives an upper bound of and for rainbow connection number where and , 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Interconnection Networks and Systems
