On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles

We give some new advances in the research of the maximum number of triangles that we may obtain in a simple arrangements of n lines or pseudo-lines.Comment: 12 pages, 9 figure

Cell-paths in mono- and bichromatic line arrangements in the plane

We show that in every arrangement of n red and blue lines | in general position and not all of the same color | there is a path through a linear number of cells where red and blue lines are crossed alternatingly (and no cell is revisited). When all lines have the same color, and hence the preceding alternating constraint is dropped, we prove that the dual graph of the arrangement always contains a path of length (n2).Peer ReviewedPostprint (author’s final draft

Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles

Flip graphs of combinatorial and geometric objects are at the heart of many deep structural insights and connections between different branches of discrete mathematics and computer science. They also provide a natural framework for the study of reconfiguration problems. We study flip graphs of arrangements of pseudolines and of arrangements of pseudocircles, which are combinatorial generalizations of lines and circles, respectively. In both cases we consider triangle flips as local transformation and prove conjectures regarding their connectivity. In the case of $n$ pseudolines we show that the connectivity of the flip graph equals its minimum degree, which is exactly $n-2$. For the proof we introduce the class of shellable line arrangements, which serve as reference objects for the construction of disjoint paths. In fact, shellable arrangements are elements of a flip graph of line arrangements which are vertices of a polytope (Felsner and Ziegler; DM 241 (2001), 301--312). This polytope forms a cluster of good connectivity in the flip graph of pseudolines. In the case of pseudocircles we show that triangle flips induce a connected flip graph on \emph{intersecting} arrangements and also on cylindrical intersecting arrangements. The result for cylindrical arrangements is used in the proof for intersecting arrangements. We also show that in both settings the diameter of the flip graph is in $\Theta(n^3)$. Our constructions make essential use of variants of the sweeping lemma for pseudocircle arrangements (Snoeyink and Hershberger; Proc.\ SoCG 1989: 354--363). We finally study cylindrical arrangements in their own right and provide new combinatorial characterizations of this class

Sweeps, arrangements and signotopes

AbstractSweeping is an important algorithmic tool in geometry. In the first part of this paper we define sweeps of arrangements and use the “Sweeping Lemma” to show that Euclidean arrangements of pseudolines can be represented by wiring diagrams and zonotopal tilings. In the second part we introduce a further representation for Euclidean arrangements of pseudolines. This representation records an “orientation” for each triple of lines. It turns out that a “triple orientation” corresponds to an arrangement exactly if it obeys a generalized transitivity law. Moreover, the “triple orientations” carry a natural order relation which induces an order relation on arrangements. A closer look on the combinatorics behind this leads to a series of signotope orders closely related to higher Bruhat orders. We investigate the structure of higher Bruhat orders and give new purely combinatorial proofs for the main structural properties. Finally, we reconnect the combinatorics of the second part to geometry. In particular, we show that the maximum chains in the higher Bruhat orders correspond to sweeps

Triangles in Euclidean Arrangements

The number of triangles in arrangements of lines and pseudolines has been object of some research. Most results, however, concern arrangements in the projective plane. In this article we add results for the number of triangles in Euclidean arrangements of pseudolines. Though the change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs. In 1926 Levi proved that a nontrivial arrangement-simple or not- of n pseudolines in the projective plane contains at least n triangles. To show the corresponding result for the Euclidean plane, namely, that a simple arrangement of n pseudolines contains at least n \Gamma 2 triangles, we had to find a completely different proof. On the other hand a non-simple arrangements of n pseudolines in the Euclidean plane can have as few as 2n=3 triangles and this bound is best possible. We also discuss the maximal possible number of triangles and som..

Triangles in euclidean arrangements

E mail ffelsnerkriegelginffuberlinde Abstract The number of triangles in arrangements of lines and pseudolines has been object of some research Most results however concern arrangements in the projective plane In this article we add results for the number of triangles in Euclidean arrange ments of pseudolines Though the change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs In Levi proved that a nontrivial arrangement simple or not of n pseudolines in the projective plane contains n triangles To show the corresponding result for the Euclidean plane namely that a simple arrangement of n pseudolines contains n triangles we had to nd a completely dierent proof On the other hand a nonsimple arrangements of n pseudolines in the Euclidean plane can have as few as n triangles and this bound is best possible We also discuss the maximal possible number of triangles and some extensions Mathematics Subject Classications A C Key Words Arrangement Euclidean plane pseudoline strechability triangl