Homotopy types of box complexes

TL;DR

This paper investigates the homotopy types of box complexes and compares various topological lower bounds for graph chromatic number, revealing their relationships and properties.

Contribution

It provides new insights into the homotopy types of box complexes and clarifies the relationships between different topological bounds for chromatic number.

Findings

Lovasz's bound can be expressed as (\u2206(G))+2

Sarkaria's bound can be expressed as (_0(G))+1

The difference between these bounds is at most 1

Abstract

In [MZ04] Matousek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovasz's original bound [L78] can be restated as $\chr G \geq \ind (\B(G)) +2$. Sarkaria's bound [S90] can be formulated as $\chr G \geq \ind (\B_0(G)) +1$. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. Some of the results was announced in [MZ04].