Box complexes, neighborhood complexes, and the chromatic number

Lovászʹs striking proof of Kneserʹs 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 Ziegler) to the Lovász complex. In the process, we analyze and clarify the combinatorics of the complexes involved and link their structure via several “intermediate” complexes.