Topological graph theory
Coloring face-hypergraphs of graphs on surfaces
Journal of Combinatorial Theory Series B
Dirac's map-color theorem for choosability
Journal of Graph Theory
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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√g)- choosable. This bound matches a previously known lower bound of order Ω(3√g). If each face of the graph is incident with at least r distinct vertices, then the face hypergraph is also O(r√g)-choosable. Note that colorings of face hypergraphs for r = 2 correspond to usual vertex colorings and the upper bound O(√g) 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 + √36g + 49/6⌉ for surfaces of Euler genus g ≥ 3. Our results dominate the previously known bounds for all genera except for g = 4, 7, 8, 9, 14.