Box complexes, neighborhood complexes, and the chromatic number

Lovasz's striking proof of Kneser's conjecture from 1978 using the Borsuk--Ulam theorem provides a lower bound on the chromatic number of a graph. 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 Z_2-deformation retraction from the box complex (in the version introduced by Matousek and Ziegler) to the Lovasz complex. In the process, we analyze and clarify the combinatorics of the complexes involved and link their structure via several ``intermediate'' complexes.Comment: 8 pages, 1 figur

Finding the Sink Takes Some Time: An Almost Quadratic Lower Bound for Findingthe Sink of Unique Sink Oriented Cubes

We give a worst-case Ω(n 2/log n) lower bound on the number of vertex evaluations a deterministic algorithm needs to perform in order to find the (unique) sink of a unique sink oriented n-dimensional cube. We consider the problem in the vertex-oracle model, introduced in [18]. In this model one can access the orientation implicitly, in each vertex evaluation an oracle discloses the orientation of the edges incident to the queried vertex. An important feature of the model is that the access is indeed arbitrary, the algorithm does not have to proceed on a directed path in a simplex-like fashion, but could "jump around”. Our result is the first superlinear lower bound on the problem. The strategy we describe works even for acyclic orientations. We also give improved lower bounds for small values of n and fast algorithms in a couple of important special classes of orientations to demonstrate the difficulty of the lower bound proble

Jumping doesn’t help in abstract cubes, in

Abstract. We construct a class of abstract objective functions on the cube, such that the algorithm BottomAntipodal takes exponentially many steps to find the maximum. A similar class of abstract objective functions is constructed for the process BottomTop, also requiring exponentially many steps

Finding the Sink Takes Some Time: An Almost Quadratic Lower Bound for Finding the Sink of Unique Sink Oriented Cubes

ISSN:0179-5376ISSN:1432-044