## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, January 24, 2008, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Philipp Zumstein

## Polychromatic Colorings of Plane Graphs

A vertex k-coloring of a plane graph G is called polychromatic if in every face of G all k colors appear. Let p(G) be the maximum number k for which there is a polychromatic k-coloring.

For a plane graph G, let g(G) denote the length of the shortest face in G. We show p(G) >= (3g(G)-5)/4 and on the other hand for each g there is a graph with g(G) = g and p(G) <= (3g+1)/4.

Furthermore, 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.

Joint work with Noga Alon, Robert Berke, Kevin Buchin, Maike Buchin, Peter Csorba, Saswata Shannigrahi, and Bettina Speckmann.

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