Rainbow connections for planar graphs and line graphs

TL;DR

This paper investigates the computational complexity of rainbow connectivity problems in various classes of graphs, proving NP-Completeness results for planar bipartite and line graphs, and providing bounds for outerplanar graphs.

Contribution

It establishes NP-Completeness of rainbow connectivity decision problems in planar bipartite and line graphs, and offers upper bounds for outerplanar graphs with small diameters.

Findings

Deciding rainbow connectivity remains NP-Complete for planar bipartite graphs.

Deciding rainbow vertex connectivity remains NP-Complete for line graphs.

Upper bounds for rainbow connection number in outerplanar graphs with small diameters.

Abstract

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. It was proved that computing $rc(G)$ is an NP-Hard problem, as well as that even deciding whether a graph has $rc(G)=2$ is NP-Complete. It is known that deciding whether a given edge-colored graph is rainbow connected is NP-Complete. We will prove that it is still NP-Complete even when the edge-colored graph is a planar bipartite graph. We also give upper bounds of the rainbow connection number of outerplanar graphs with small diameters. A vertex-colored graph is rainbow vertex-connected if any two vertices 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. It is known that deciding whether a given vertex-colored graph is rainbow vertex-connected is NP-Complete. We will prove that it is still NP-Complete even when the vertex-colored graph is a line graph.