Florian Zickfeld

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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)
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)
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)
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)
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)
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 memory of Torsten Küther. iii ACKNOWLEDGEMENTS I want to thank Professor Yu for finding interesting problems and giving me the opportunity to work on them. Not only was he willing to give good general advice, but also discuss specific details, even in the write-up itself. Of course I want to thank the Fulbright Program and the German National Academic(More)
New lower bounds for the number of (≤ k)-edges and the rectilinear crossing number of K n A characterization of the faces of the generalized cluster complex ∆ m (Φ) Abstract Given an integer matrix B, the Chvatal rank of B is the minimum number k such that for every right-hand side b, the integer hull of the rational polyhedron {x : Bx ≤ b} can be obtained(More)