Rainbow connections for outerplanar graphs with diameter 2 and 3

An edge-colored graph G is rainbow connected if every two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rcðGÞ, is the smallest number of colors that are needed in order to make G rainbow connected. It was proved that computing rcðGÞ is an NP-hard problem, as well as that even deciding whether a graph has rcðGÞ 1⁄4 2 is NP-complete. Li et al. proved that rcðGÞ 6 5 if G is a bridgeless graph with diameter 2, while rcðGÞ 6 9 if G is a bridgeless graph with diameter 3. Furthermore, Uchizawa et al. showed that determining the rainbow connection number of graphs is strongly NP-complete even for outerplanar graphs. In this paper, we give upper bounds of the rainbow connection number of outerplanar graphs with small diameters. 2014 Elsevier Inc. All rights reserved.