 
Summary: Polychromatic Colorings of Plane Graphs
Noga Alon
Robert Berke
Kevin Buchin
Maike Buchin
P´eter Csorba §
Saswata Shannigrahi ¶
Bettina Speckmann §
Philipp Zumstein
November 12, 2007
Abstract
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 NPcomplete
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.
1 Introduction
