Rainbow connection number and graph operations

Abstract

A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. An edge-colored graph G is rainbow connected if every two distinct vertices are connected by a rainbow path. The rainbow connection numberrc(G) of G is the smallest number of colors that are needed in order to make G rainbow connected. In this paper, we study bounds of rainbow connection number of some graph operations, such as the union of two graphs, adding edges, deleting edges, and adding vertices and edges. Moreover, we also study the following extremal problem. Let k and n be two integers such that 1≤k≤ℓ<n. Find the smallest integer f(n,k,ℓ) such that for each graph G of order n and diameter k, there exists an edge set F⊆E(G¯) satisfying |F|≤f(n,k,ℓ) and rc(G+F)≤ℓ.