Existence and Construction of Schnyder Orientations over a Large Class of Higher Genus Surface Triangulations
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Given a planar triangulation, a Schnyder wood is an orientation and coloring of the edges of the triangulation which respect a particular type of local symmetry around each of the vertices. Schnyder woods are well studied objects in the theory of planar cellular graph embeddings, and in particular their characterization and existence properties are well known. A Schnyder orientation is a recently developed generalization of a Schnyder wood to higher genus orientable surfaces for which characterization and existence results are only known in the case of toroidal triangulations. In this thesis we develop a construction method for Schnyder orientations on higher genus surface triangulations, as well as a structural theory that allows us to prove their existence on all triangulations on a genus $g \ge 1$ orientable surface provided that the triangulations have edge-width at least $40(2^g - 1)$. At this time, this is the only existence result for Schnyder orientations over a relatively large class of triangulations with a supporting surface of genus $g \ge 2$.
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