Polychromatic colorings of plane graphs

We show that the vertices of any plane graph in which every face is of length at least g can be colored by (3g - 5)/4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g + 1)/4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of length 3 or 4 only. If all faces are of length 3 this can be decided in polynomial time. The investigation of this problem is motivated by its connection to a variant of the art gallery problem in computational geometry.

• Type: Conference
• Year: 2008
• Where: COMPGEOM
• Authors: Noga Alon, Robert Berke, Kevin Buchin, Maike Buchin, Péter Csorba, Saswata Shannigrahi, Bettina Speckmann, Philipp Zumstein
• Keywords: Color, COMPGEOM 2008, Discrete Geometry, Plane Graph, Vertex Coloring