Topological graph theory
Neighborly maps with few vertices
Discrete & Computational Geometry
Neighborly 2-manifolds with 12 vertices
Journal of Combinatorial Theory Series A
Journal of Combinatorial Theory Series B
Construction and representation of neighborly manifolds
Journal of Combinatorial Theory Series A
Exponential families of non-isomorphic triangulations of complete graphs
Journal of Combinatorial Theory Series B
On the number of nonisomorphic orientable regular embeddings of complete graphs
Journal of Combinatorial Theory Series B
Nonisomorphic complete triangulations of a surface
Discrete Mathematics
On the Folkman-Lawrence topological representation theorem for oriented matroids of rank 3
European Journal of Combinatorics - Special issue on combinatorial geometries
Computational Oriented Matroids
Computational Oriented Matroids
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We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g+1 curves in contrast to the non-orientable surface where the number of curves is not bounded.