## Journée "Flips" (28 novembre 2017)

_{FLIPS}

mardi 28 novembre 2017

Cette journée sera consacrée aux problèmes de flips. Elle s'inscrit dans le cadre de l'axe 3 (Physique Statistique, Combinatoire) du pôle Math-STIC de l'Université Paris 13, qui fédère les laboratoires de mathématiques (LAGA), d'informatique (LIPN) et de traitement et transmission de l'information (L2TI).

La conférence a lieu en salle B107 du LIPN, voir infos pratiques ci-dessous.

**Programme**

10h: **Alexander Pilz** (ETH Zürich) : *Determining the flip distance between triangulations of simple polygons and point sets*

In this talk, we consider edge exchange flips in triangulations of point sets and simple polygons. In particular, it addresses the question whether the length of the shortest sequence of flips for transforming one given triangulation into another can be determined in polynomial time. We present a proof showing that the corresponding optimization problem is APX-hard for triangulations of point sets. For triangulations of simple polygons, we give a reduction showing NP-completeness of the decision version of the problem. The two proofs are fundamentally different, but use a common sub-structure, the well-known double chain, whose properties with respect to flip graphs we investigate. Finally, we point out some difficulties in solving the flip distance problem for triangulations of convex polygons, which is still open. Based in part on joint work with Oswin Aichholzer and Wolfgang Mulzer.

11h15: **Thomas Budzinski** (École Normale Supérieure) : *On the mixing time of the flip walk on triangulations of the sphere*

A simple way to sample a uniform triangulation of the sphere with a fixed number $n$ of vertices is a Monte-Carlo method: we start from an arbitrary triangulation and flip repeatedly a uniformly chosen edge. We prove a lower bound of order n^(5/4) on the mixing time of this Markov chain.

12h30: Lunch

14h: **Lionel Pournin** (LIPN, Université Paris 13) : *Eccentricities in the flip-graphs of convex polygons*

The flip-graph of a convex polygon π is the graph whose vertices are the triangulations of π and whose edges correspond to flips between them. The eccentricity of a triangulation T of π is the largest possible distance in this graph from T to any triangulation of π. Let n stand for the number of vertices of π. It is well known that, when all n-3 interior edges of T are incident to a given vertex, the eccentricity of T in the flip-graph of π is exactly n-3. The purpose of this talk is to generalize this statement to arbitrary triangulations of π: if n-3-k denotes the largest number of interior edges of T incident to a vertex, and if k≤n/2-2, the eccentricity of T in the flip-graph of π is exactly n-3+k. Surprisingly, the value of k turns out to characterize eccentricities if it is small enough. More precisely, when k≤n/8-5/2, T has eccentricity n-3+k if and only if exactly n-3-k of its interior edges are incident to a given vertex. A number of related questions will be mentioned and discussed.

15h15: **Thomas Fernique** (LIPN, Université Paris 13) : *Flips in Tilings*

Tilings are rigid geometric structures used, for example, to model materials. A flip on tilings is a local operation which changes a tiling into another one, yielding a graph whose nodes are the different tilings and edges connect tilings which differ by a flip. A well-known example are tilings by dominos (1x2 and 2x1 rectangles), where a flip consists in a quarter turn on a 2x2 square tiled by two dominos.

Issues as connectedness of diameter of such graphs (as well as more advanced issues as mixing time of random walks) are of importance to better understand the physical properties of the modeled material. The aim of this talk is to review several examples of flips on tilings and the corresponding results or conjectures.

**Aspects pratiques**

Les exposés se tiendront en salle B107 (Bâtiment B, 1er étage) de l'institut Galilée, Université Paris 13. Cliquer ici pour des plans et moyens d'accès.

Contact : Lionel Pournin <This email address is being protected from spambots. You need JavaScript enabled to view it.