Rainbow connectivity of randomly perturbed graphs

Abstract

In 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 .

Rainbow connections of bioriented graphs

“…Rainbow connectivity finds significant applications in network security, attracting the attention of numerous scholars and yielding substantial results for both general graphs and digraphs. Furthermore recently some new results ( [11] , [10] , [12] ) which are concerning the rainbow connectivity in some random graph models are obtained.…”