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 $${{\,\textrm{src}\,}}(G)={{\,\textrm{diam}\,}}(G)$$ .