Rainbow connection of sparse random graphs
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 \(r=3\).
scientific article; zbMATH DE number 6171788