Higher connectivity of graph coloring complexes

TL;DR

This paper proves a conjecture relating the connectivity of graph coloring complexes to the maximum degree of the graph, establishing new lower bounds for their topological connectivity.

Contribution

It provides a proof of Babson & Kozlov's conjecture, showing that Hom(G,K_n) complexes are at least (n-d-2)-connected, and derives specific results for odd cycles.

Findings

Hom(G,K_n) is at least (n-d-2)-connected for graphs with maximum degree d

Hom(C_{2r+1},K_n) is (n-4)-connected for n ≥ 3

Topological bounds for graph chromatic numbers are improved

Abstract

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,K_n) is at least (n-d-2)-connected. Here Hom(-,-) denotes the polyhedral complex introduced by Lov\'asz to study the topological lower bounds for chromatic numbers of graphs. We will also prove, as a corollary to the main theorem, that the complex Hom(C_{2r+1},K_n) is (n-4)-connected, for $n\geq 3$.