ar X iv : 0 71 1 . 28 49 v 1 [ m at h . C O ] 1 9 N ov 2 00 7 Partitioning complete graphs by heterochromatic trees ∗

ar X iv : 0 71 1 . 28 84 v 1 [ m at h . R T ] 1 9 N ov 2 00 7 UNIQUENESS AND DISJOINTNESS OF KLYACHKO MODELS

We show the uniqueness and disjointness of Klyachko models for GLn over a non-archimedean local field. This completes, in particular, the study of Klyachko models on the unitary dual. Our local results imply a global rigidity property for the discrete automorphic spectrum.

ar X iv : 0 71 1 . 28 46 v 1 [ m at h . C O ] 1 9 N ov 2 00 7 Rainbow number of matchings in regular bipartite graphs ∗

Given a graph G and a subgraph H of G, let rb(G,H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G,H) is called the rainbow number of H with respect to G. Denote mK2 a matching of size m and Bn,k a k-regular bipartite graph with bipartition (X,Y ) such that |X| = |Y | = n and k ≤ n. In this paper we give an upper and lower bound f...

ar X iv : 0 71 1 . 23 72 v 1 [ m at h . G R ] 1 5 N ov 2 00 7 Braid groups and Artin groups

ar X iv : m at h / 06 11 62 6 v 1 [ m at h . C O ] 2 1 N ov 2 00 6 COUNTING LINKS IN COMPLETE GRAPHS

We find the minimal number of links in an embedding of any complete k-partite graph on 7 vertices (including K 7 , which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete k-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with min...

ar X iv : 0 71 1 . 15 17 v 1 [ m at h . C O ] 9 N ov 2 00 7 COMBINATORIAL POLAR ORDERINGS AND FOLLOW - UP ARRANGEMENTS

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal comp...