The rainbow connected number of several infinite graph families

Abstract

Let G be a nontrivial connected graph with an edge coloring, and let \(u,v \in V(G)\). A \(u-v\) path in G is said to be a rainbow path if no color is repeated on the edges of the path. Similarly, we define a rainbow geodesic. A rainbow connected graph G is a graph with an edge coloring such that every two vertices in G are connected by a rainbow path. Further, a strong rainbow connected graph G is a graph with an edge coloring such that every two vertices in G is connected by a rainbow geodesic. The minimum number of colors needed to make a graph rainbow connected is called the rainbow connection number, denoted \({{\,\textrm{rc}\,}}(G)\), and the minimum number of colors needed to make a graph strong rainbow connected is called the strong rainbow connection number, denoted \({{\,\textrm{src}\,}}(G)\). In this paper we determine \({{\,\textrm{rc}\,}}(G)\) and \({{\,\textrm{src}\,}}(G)\) when G is a n-dimensional rectangular grid graph, triangular grid graph, hexagonal grid graph, and a (weak) Bruhat graph, respectively. We show for all these families that \({{\,\textrm{src}\,}}(G)={{\,\textrm{diam}\,}}(G)\).