Note on the Hardness of Rainbow Connections for Planar and Line Graphs

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.

Rainbow connections for outerplanar graphs with diameter 2 and 3

“…In [11], the authors proved that it is still NP-complete even when the edge-colored graph is a planar bipartite graph. Uchizawa et al [22] obtained a stronger result: Determining the rainbow connection number of graphs is strongly NP-complete even for outerplanar graphs.…”

“…Actually, it is already NP-complete to decide whether rcðGÞ ¼ 2, and in fact it is already NP-complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. In [11], we proved that it is still NP-complete even when the edge-colored graph is a planar bipartite graph. Uchizawa et al [22] obtained a stronger result: Determining the rainbow connection number of graphs is strongly NP-complete even for outerplanar graphs.…”

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

“…Building on their result, Li et al [87] proved Rainbow Connectivity remains NP-complete for bipartite graphs. Furthermore, the problem is NP-complete even for bipartite planar graphs as shown by Huang et al [63]. Recently, Uchizawa et al [145] complemented these results by showing Rainbow Connectivity is NP-complete for outerplanar graphs, and even for series-parallel graphs.…”

Injective coloring of some graph operations

“…Excluding classical vertex coloring and edge coloring, many types of coloring are studied, such as list coloring, acyclic coloring, and star coloring. In addition, rainbow connection and rainbow vertex-connection are new types of coloring, as described in a recent survey [12] and other studies [2,6,7,[9][10][11]. In the present study, we consider injective coloring.…”