Rainbow connection and strong rainbow connection of the crystal graph and neurons graph

Abstract

A rainbow path in a graph G is an edge colored graph such that each edge in the path does not have the same color. The minimal number of colors which is needed to color the edges in the path so that every pair of vertices is connected by at least one rainbow path is called the rainbow connection number of G, denoted by rc(G). A rainbow path of length d(u, v), where d(u, v) is the distance between u and v, is called rainbow u - v geodesic in G. A strongly rainbow connected graph is a graph such that there exists a rainbow u-v geodesic for any two vertices u and v in G. The strong rainbow connection number src(G) of G is the minimum number of colors needed to make G strongly rainbow connected graph. In this paper, we prove the exact values of rc and src for crystal graph and its generalization to neuron graph.