Strong rainbow connection numbers of helm graphs and amalgamation of lollipop graphs

Abstract

An edge coloring of a graph is called rainbow coloring if any pair of vertices are connected by a path with different colors of edges. Let G be a simple connected graph. A rainbow u-v geodesic in G is a path between two vertices u and v in G of length d(u,v), where d(u,v) is the distance of those vertices, and all of its edges are colored by different colors. If G contains a rainbow u-v geodesic for every pair of distinct vertices u and v in this graph, then G is called strongly rainbow connected graph. The strong rainbow connection number of G, denoted by src(G), is the least number of colors needed to make Gstrongly rainbow connected graph. In this paper, we prove that helm graphs and amalgamation of lollipop graphs are strongly rainbow connected graphs.