Publication details
Coloring face hypergraphs on surfaces
| Authors | |
|---|---|
| Year of publication | 2005 |
| Type | Article in Periodical |
| Magazine / Source | European Journal of Combinatorics |
| Citation | |
| Doi | http://dx.doi.org/10.1016/j.ejc.2004.01.003 |
| 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. |