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It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4 + 2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K 4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called… (More)

In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of… (More)

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with out-degrees prescribed by a function α : V → N unifies many different combinatorial structures, including the afore… (More)

Acknowledgments First of all I want to thank my advisor, Stefan Felsner. In lectures and many discussions I learned a lot from him not only about graph theory and combinatorics. He was also willing to share the large and little tricks and insights that make (scientific) working so much easier. During lunches and many coffee breaks I also learned a lot from… (More)

We deal with the enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect match-ings and ice models. The notion of an α-orientation unifies many different combinatorial structures, including the afore mentioned. We ask for the number of α-orientations and also for… (More)

Hajós conjectured that, for any positive integer k, every graph containing no K k+1-subdivision is k-colorable. This is true when k ≤ 3, and false when k ≥ 6. Hajós' conjecture remains open for k = 4, 5. In this paper, we show that any possible counterexample to this conjecture for k = 4 with minimum number of vertices must be 4-connected. This is a step in… (More)

In the last six months I have mainly worked on problems from two different areas. I will first give an overview of my work on the number of orientations with fixed out-degrees. In contrast to this the second topic is rather concerned with structural graphs properties, namely spanning trees with many leaves. I have worked on the problems presented in this… (More)