WS 06 / 07 of Florian Zickfeld

I have worked on the problems presented in this section with Stefan Felsner. A paper [8] is a available on the arXiv and a conference version [9] has been submitted. I will now outline what type of questions we consider and our main results. The concept of orientations with fixed out-degrees or α-orientations is a quite general one as we will see below. Let a graph G with vertex set V and a function α : V → N be given. An orientation X of the edges of G is an α-orientation if every vertex v has out-degree α(v). A planar map is a planar graph together with a plane drawing. Felsner [6] proves that for fixed α and M the set of α-orientations of a planar map M is a distributive lattice. This structure on the set of α-orientations found applications in drawing algorithms for example in [2], and for enumeration and random sampling of graphs in [10]. We also focus on the study of α-orientations of planar maps in the sequel. Felsner [6] describes encodings as α-orientations for Schnyder woods on 3-connected planar maps, bipartite f -factors (especially bipartite perfect matchings), and spanning trees. Furthermore, Eulerian orientations, bipolar orientations, and 2-orientations of quadrangulations can all be encoded as α-orientations. For some of the above mentioned structures it is not obvious how to encode them as αorientations. For Schnyder woods on triangulations the encoding by 3-orientations goes back to de Fraysseix and de Mendez [5]. For bipolar orientations an encoding was proposed by D. Woods and independently by Tamassia and Tollis [16]. Bipolar orientations of M are one of the structures which cannot be encoded as α-orientations on M , an auxiliary map M ′ (the angle graph of M) has to be used instead. Given the existence of a combinatorial structure on the elements of a class Mn of planar maps with n vertices, one of the questions of interest is how many such structures there are for a given map M ∈ Mn. Especially, one is interested in the minimum and maximum that this number attains on the maps from Mn. This question has been treated quite successfully for spanning trees and bipartite perfect matchings. For spanning trees the Kirchhoff Matrix Tree Theorem comes into the game and allows to bound the maximum number of spanning trees of a planar graph with n vertices between 5.02 and 5.34, see [15]. Pfaffian orientations