Rainbow cycles in flip graphs

The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon, and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly $r$~times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of $r$-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex $n$-gon, the flip graph of plane trees on an arbitrary set of $n$~points, and the flip graph of non-crossing perfect matchings on a set of $n$~points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of $\{1,2,\dots,n\}$ and the flip graph of $k$-element subsets of $\{1,2,\dots,n\}$. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of~$r$, $n$ and~$k$

An efficient sampling algorithm for difficult tree pairs

It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees. The problem of computing the rotation distance between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees (S', T'), where there is no known first step which is guaranteed to be the beginning of a minimal length path. Of interest, therefore, is how to sample such difficult pairs of trees of a fixed size. We show that it is possible to do so efficiently, and present such an algorithm that runs in time O(n4)

Structural properties of non-crossing partitions from algebraic and geometric perspectives

The present thesis studies structural properties of non-crossing partitions associated to finite Coxeter groups from both algebraic and geometric perspectives. On the one hand, non-crossing partitions are lattices, and on the other hand, we can view them as simplicial complexes by considering their order complexes. We make use of these different interpretations and their interactions in various ways. The order complexes of non-crossing partitions have a rich geometric structure, which we investigate in this thesis. In particular, we interpret them as subcomplexes of spherical buildings. From a more algebraic viewpoint, we study automorphisms and anti-automorphisms of non-crossing partitions and their relation to building automorphisms. We also compute the automorphism groups of non-crossing partitions of type $B$ and $D$, provided that $n \neq 4$ for type $D$. For this, we introduce a new pictorial representation for type $D$. In type $A$ we study the structural properties of the order complex of the non-crossing partitions in more detail. In particular, we investigate the interaction of chamber distances and convex hulls in the non-crossing partition complex and the ambient spherical building. These questions are connected to the curvature conjecture of Brady and McCammond.Comment: 181 pages, doctoral dissertation, Karlsruher Instituts f\"ur Technologie (2018