Publication details

Coloring face hypergraphs on surfaces



Year of publication 2005
Type Article in Periodical
Magazine / Source European Journal of Combinatorics
Description The face hypergraph of a graph G embedded on a surface has the same vertex set as G and its edges are the sets of vertices forming faces of G. A hypergraph is k-choosable if for each assignment of lists of colors of sizes k to its vertices, there is a coloring of the vertices from these lists avoiding a monochromatic edge. We prove that the face hypergraph of the triangulation of a surface of Euler genus g is O((3)rootg)-choosable. This bound matches a previously known lower bound of order Omega((3)rootg). If each face of the graph is incident with at least r distinct vertices, then the face hypergraph is also O( (r)rootg)-choosable. Note that colorings of face hypergraphs for r = 2 correspond to usual vertex colorings and the upper bound O(rootg) thus follows from Heawood's formula. Separate results for small genera are presented: the bound 3 for triangulations of the surface of Euler genus g = 3 and the bound [7 + root36g + 49/6] for 6 surfaces of Euler genus g greater than or equal to 3. Our results dominate the previously known bounds for all genera except for g = 4, 7, 8, 9. 14. (C) 2004 Elsevier Ltd. All rights reserved.

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