Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs

We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, "Counting labelled projective-planar graphs without a K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table

Hamiltonian cycles in maximal planar graphs and planar triangulations

In this thesis we study planar graphs, in particular, maximal planar graphs and general planar triangulations. In Chapter 1 we present the terminology and notations that will be used throughout the thesis and review some elementary results on graphs that we shall need. In Chapter 2 we study the fundamentals of planarity, since it is the cornerstone of this thesis. We begin with the famous Euler's Formula which will be used in many of our results. Then we discuss another famous theorem in graph theory, the Four Colour Theorem. Lastly, we discuss Kuratowski's Theorem, which gives a characterization of planar graphs. In Chapter 3 we discuss general properties of a maximal planar graph, G particularly concerning connectivity. First we discuss maximal planar graphs with minimum degree i, for i = 3; 4; 5, and the subgraph induced by the vertices of G with the same degree. Finally we discuss the connectivity of G, a maximal planar graph with minimum degree i. Chapter 4 will be devoted to Hamiltonian cycles in maximal planar graphs. We discuss the existence of Hamiltonian cycles in maximal planar graphs. Whitney proved that any maximal planar graph without a separating triangle is Hamiltonian, where a separating triangle is a triangle such that its removal disconnects the graph. Chen then extended Whitney's results and allowed for one separating triangle and showed that the graph is still Hamiltonian. Helden also extended Chen's result and allowed for two separating triangles and showed that the graph is still Hamiltonian. G. Helden and O. Vieten went further and allowed for three separating triangles and showed that the graph is still Hamiltonian. In the second section we discuss the question by Hakimi and Schmeichel: what is the number of cycles of length p that a maximal planar graph on n vertices could have in terms of n? Then in the last section we discuss the question by Hakimi, Schmeichel and Thomassen: what is the minimum number of Hamiltonian cycles that a maximal planar graph on n vertices could have, in terms of n? In Chapter 5, we look at general planar triangulations. Note that every maximal planar graph on n ≥ 3 vertices is a planar triangulation. In the first section we discuss general properties of planar triangulations and then end with Hamiltonian cycles in planar triangulations