Topological graph theory
The number of cycles in 2-factors of cubic graphs
Discrete Mathematics
Five-coloring maps on surfaces
Journal of Combinatorial Theory Series B
List colourings of planar graphs
Discrete Mathematics
Coloring graphs without short non-bounding cycles
Journal of Combinatorial Theory Series B
Every planar graph is 5-choosable
Journal of Combinatorial Theory Series B
Handbook of combinatorics (vol. 1)
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Color-critical graphs on a fixed surface
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Face 2-colourable triangular embeddings of complete graphs
Journal of Combinatorial Theory Series B
Discrete Mathematics
Exponential families of non-isomorphic triangulations of complete graphs
Journal of Combinatorial Theory Series B
Colouring Eulerian triangulations
Journal of Combinatorial Theory Series B
Coloring locally bipartite graphs on surfaces
Journal of Combinatorial Theory Series B
Maximum genus embeddings of Steiner triple systems
European Journal of Combinatorics - Special issue: Topological graph theory II
Graphs and Hypergraphs
Graph Theory With Applications
Graph Theory With Applications
Dirac's map-color theorem for choosability
Journal of Graph Theory
Discrete Mathematics - Kleitman and combinatorics: a celebration
Coloring face hypergraphs on surfaces
European Journal of Combinatorics
2-List-coloring planar graphs without monochromatic triangles
Journal of Combinatorial Theory Series B
Hi-index | 0.00 |
The face-hypergraph, H(G), of a graph G embedded in a surface has vertex set V(G), and every face of G corresponds to an edge of H(G) consisting of the vertices incident to the face. We study coloring parameters of these embedded hypergraphs. A hypergraph is k-colorable (k-choosable) if there is a coloring of its vertices from a set of k colors (from every assignment of lists of size k to its vertices) such that no edge is monochromatic. Thus a proper coloring of a face-hypergraph corresponds to a vertex coloring of the underlying graph such that no face is monochromatic. We show that hypergraphs can be extended to face-hypergraphs in a natural way and use tools from topological graph theory, the theory of hypergraphs, and design theory to obtain general bounds for the coloring and choosability problems. To show the sharpness of several bounds, we construct for every even n an n-vertex pseudo-triangulation of a surface such that every triple is a face exactly once. We provide supporting evidence for our conjecture that every plane face-hypergraph is 2-choosable and we pose several open questions, most notably: Can the vertices of a planar graph always be properly colored from lists of size 4, with the restriction on the lists that the colors come in pairs and a color is in a list if and only if its twin color is? An affirmative answer to this question would imply our conjecture, as well as the Four Color Theorem and several open problems.