Rainbow connectivity of randomly perturbed graphs

Abstract

AbstractIn this note we examine the following random graph model: for an arbitrary graph , with quadratic many edges, construct a graph by randomly adding edges to and randomly coloring the edges of with colors. We show that for a large enough constant and , every pair of vertices in are joined by a rainbow path, that is, is rainbow connected, with high probability. This confirms a conjecture of Anastos and Frieze, who proved the statement for and resolved the case when and is a function of .