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

An edge-colored graph $$G$$G is rainbow connected if every two vertices are connected by a path whose edges have distinct colors. 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. A vertex-colored graph is rainbow vertex-connected if every two vertices are connected by a path whose internal vertices have distinct colors. 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.

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