Two Results in Drawing Graphs on Surfaces

In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian and their structure allows restoration of edge-hamiltonicity with only a small change

Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

Nowhere-zero flows and structures in cubic graphs

Wir widerlegen zwei Vermutungen, die im Zusammenhang mit Kreisüberdeckungen von kubischen Graphen stehen. Die erste Vermutung, welche kubische Graphen mit dominierenden Kreisen betrifft, widerlegen wir durch Erweiterung eines Theorems von Gallai über induzierte eulersche Graphen und durch Konstruktion spezieller snarks. Die zweite Vermutung, welche frames betrifft, widerlegen wir durch Betrachtung der Frage nach der Existenz von speziellen spannenden Teilgraphen in 3-fach zusammenhängenden kubischen Graphen. Weiters übersetzen wir Probleme über Flüsse in kubischen Graphen in Knotenfärbungsprobleme von planaren Graphen und erhalten eine neue Charakterisierung von snarks. Schliesslich verbessern und erweitern wir Resultate über Knotenfärbungsprobleme in Quadrangulierungen. Zu Ende stellen wir neue Vermutungen auf, die im Zusammenhang mit Kreisüberdeckungen und Strukturen in kubischen Graphen stehen.We disprove two conjectures which are related to cycle double cover problems. The first conjecture concerns cubic graphs with dominating cycle. We disprove this conjecture by extending a result of Gallai about induced eulerian subgraphs and by constructing special snarks. The second conjecture concerns frames. We show that this conjecture is false by considering the problem whether every 3-connected cubic graph has a spanning subgraph with certain properties. Moreover, we transform flow-problems of cubic graphs into vertex coloring problems of plane graphs. We obtain thereby a new characterization of snarks. Furthermore, we improve and extend results about vertex coloring problems of quadrangulations. Finally we pose new problems and state conjectures which are related to cycle double covers and structures in cubic graphs

Duotone Surfaces: Division of a Closed Surface into Exactly Two Regions

In this thesis work, our main motivation is to create computer aided art work which can eventually transform into a sculpting tool. The work was inspired after Taubin’s work on constructing Hamiltonian triangle strips on quadrilateral meshes. We present an algorithm that can divide a closed 2-manifold surface into exactly two regions, bounded from each other by a single continuous curve. We show that this kind of surface division is possible only if the mesh approximation of a given object is a two colorable quadrilateral mesh. For such a quadrilateral mesh, appropriate texturing of the faces of the mesh using a pair of Truchet tiles will give us a Duotone Surface. Catmull-Clark subdivision can convert any given mesh with arbitrary sided polygons into a two colorable quadrilateral mesh. Using such vertex insertion schemes, we modify the mesh and classify the vertices of the new mesh into two sets. By appropriately texturing each face of the mesh such that the color of the vertices of the face match with the colored regions of the corresponding Truchet tile, we can get a continuous curve that splits the surface of the mesh into two regions. Now, coloring the thus obtained two regions with two different colors gives us a Duotone Surface