Rainbow connection of sparse random graphs
type: Item
instance of: scholarly article
title: Rainbow connection of sparse random graphs
zbMATH Open document ID: 1266.05030
author: Charalampos E. Tsourakakis
author: Alan Frieze
published in: The Electronic Journal of Combinatorics
publication date: 2013-06-07T00:00:00Z
full work available at URL: v19i4p5
review text: Summary: An edge colored graph \(G\) is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph \(G\), denoted by \(rc(G)\), is the smallest number of colors that are needed in order to make \(G\) rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold \(p=\frac{\log n+\omega}{n}\) where \(\omega = \omega(n) \to \infty\) and \({\omega}=o(\log{n})\) and of random \(r\)-regular graphs where \(r \geq 3\) is a fixed integer. Specifically, we prove that the rainbow connectivity \(rc(G)\) of \(G=G(n,p)\) satisfies \(rc(G) \sim \max\{Z_1,\text{diam}(G)\}\) with high probability (whp). Here \(Z_1\) is the number of vertices in \(G\) whose degree equals 1 and the diameter of \(G\) is asymptotically equal to \(\frac{\log n}{\log\log n}\) whp. Finally, we prove that the rainbow connectivity \(rc(G)\) of the random \(r\)-regular graph \(G=G(n,r)\) whp satisfies \(rc(G) =O(\log^{2\theta_r}{n})\) where \(\theta_r=\frac{\log (r-1)}{\log (r-2)}\) when \(r \geq 4\) and \(rc(G) = O(\log^4n)\) whp when \(r=3\).
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C40
Mathematics Subject Classification ID: 05C80
Mathematics Subject Classification ID: 05C42
zbMATH DE Number: 6171788
zbMATH Keywords: random graphs
zbMATH Keywords: rainbow connection
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6171788