Note on the complexity of deciding the rainbow (vertex-) connectedness for bipartite graphs

Abstract

A path in an edge-colored graph is said to be a rainbow path if no two edges on the path share the same color. An edge-colored graph is (strongly) rainbow connected if there exists a rainbow (geodesic) path between every pair of vertices. The (strong) rainbow connection number of G, denoted by (scr(G), respectively) rc(G), is the smallest number of colors that are needed in order to make G (strongly) rainbow connected. A vertex-colored graph G is rainbow vertex-connected if any pair of vertices in G are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. Though for a general graph G it is NP-Complete to decide whether rc(G)=2 (or rvc(G)=2), in this paper, we show that the problem becomes easy when G is a bipartite graph. Whereas deciding whether rc(G)=3 (or rvc(G)=3) is still NP-Complete, even when G is a bipartite graph. Moreover, it is known that deciding whether a given edge(vertex)-colored (with an unbound number of colors) graph is rainbow (vertex-) connected is NP-Complete. We will prove that it is still NP-Complete even when the edge(vertex)-colored graph is bipartite. We also show that a few NP-hard problems on rainbow connection are indeed NP-Complete.