On the rainbow connection number of triangle-net graph

Abstract

Let G be an arbitrary non-trivial connected graph. For every two vertices u and v in G, a (u,v)-path in G is called a rainbow (u,v)-path if all edges are colored differently. Next, a rainbow (u, v)-geodesic in G is a rainbow (u,v)-path of length d(u,v). Graph G is rainbow connected if for every two vertices u,v ∈ V(G), there exists a rainbow (u,v)-path. If there exists a rainbow (u, v)-geodesic in G for every two vertices u, v ∈ V(G) then G is strongly rainbow connected. The rainbow connection number rc(G) is the minimum number of colors needed to make G rainbow connected, while the strong rainbow connection number src(G) is the minimum number of colors needed to make G strongly rainbow connected. Let Trn be the generalized triangle-ladder graph for n ≥ 2. The triangle-net graph, denoted by H = (Trn)m, is constructed by taking m homogeneous generalized triangle-ladder graphs and identifying their terminal vertices, for m ≥ 2. This paper determined the rainbow connection number of the triangle-net graph and the upper bound of the strong rainbow connection number of the graph.