Oriented matroids and complete-graph embeddings on surfaces

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. © 2006 Elsevier Inc. All rights reserved. E-mail addresses: [email protected] (J. Bokowski), [email protected] (T. Pisanski). 1 Joint position at the Primorska Institute of Science and Technology, University of Primorska, Koper, Slovenia. Part of the research was conducted while the author was Neil R. Grabois Visiting Professor of Mathematics at Colgate University. The research was supported in part by a grant from Ministrstvo za znanost in tehnologijo Republike Slovenije. 0097-3165/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcta.2006.06.012 2 J. Bokowski, T. Pisanski / Journal of Combinatorial Theory, Series A 114 (2007) 1–19