ar X iv : m at h / 04 10 33 5 v 1 [ m at h . C O ] 1 4 O ct 2 00 4 HIGHER CONNECTIVITY OF GRAPH COLORING COMPLEXES SONJA

ar X iv : m at h / 04 10 33 5 v 2 [ m at h . C O ] 3 1 Ja n 20 05 HIGHER CONNECTIVITY OF GRAPH COLORING COMPLEXES

The main result of this paper is a proof of the following conjecture of Babson & Kozlov: Theorem. Let G be a graph of maximal valency d, then the complex Hom (G, Kn) is at least (n − d − 2)-connected. Here Hom (−,−) denotes the polyhedral complex introduced by Lovász to study the topological lower bounds for chromatic numbers of graphs. We will also prove, as a corollary to the main theorem, th...

ar X iv : m at h / 04 10 21 7 v 1 [ m at h . C O ] 8 O ct 2 00 4 Joints in graphs

In 1969 Erd˝ os proved that if r ≥ 2 and n > n0 (r) , every graph G of order n and e (G) > tr (n) has an edge that is contained in at least n r−1 / (10r) 6r (r + 1)-cliques. In this note we improve this bound to n r−1 /r r+5. We also prove a corresponding stability result.

ar X iv : m at h / 03 10 33 9 v 1 [ m at h . C O ] 2 1 O ct 2 00 3 Box complexes , neighborhood complexes , and the chromatic number

Lovász’s striking proof of Kneser’s conjecture from 1978 using the Borsuk–Ulam theorem provides a lower bound on the chromatic number χ(G) of a graph G. We introduce the shore subdivision of simplicial complexes and use it to show an upper bound to this topological lower bound and to construct a strong Z2-deformation retraction from the box complex (in the version introduced by Matoušek and Zie...

ar X iv : h ep - t h / 04 10 18 8 v 1 18 O ct 2 00 4 Spacetime quantization induced by axial currents

ar X iv : m at h / 04 10 34 7 v 1 [ m at h . C O ] 1 5 O ct 2 00 4 COMPLETING A k − 1 ASSIGNMENT

We consider the distribution of the value of the optimal k-assignment in an m× n-matrix, where the entries are independent exponential random variables with arbitrary rates. We give closed formulas for both the Laplace transform of this random variable and for its expected value under the condition that there is a zero-cost k − 1-assignment.