Simple $S_r$-homotopy types of Hom complexes and box complexes associated to $r$-graphs

TL;DR

This paper compares two topological complexes associated with r-graphs, showing they share the same simple symmetric group homotopy type, which advances understanding of graph coloring bounds.

Contribution

It establishes that the Hom complex and the box complex for r-graphs are of the same simple S_r-homotopy type, extending known results to hypergraphs.

Findings

Hom complex and box complex are of the same simple S_r-homotopy type.

The comparison extends topological bounds from graphs to hypergraphs.

Provides tools for estimating chromatic numbers of r-graphs.

Abstract

For a pair $(H_1,H_2)$ of graphs, Lov\'{a}sz introduced a polytopal complex called the Hom complex $\text{Hom}(H_1,H_2)$, in order to estimate topological lower bounds for chromatic numbers of graphs. The definition is generalized to hypergraphs. Denoted by $K_r^r$ the complete $r$-graph on $r$ vertices. Given an $r$-graph $H$, we compare $\text{Hom}(K_r^r,H)$ with the box complex $\mathsf{B}_{\text{edge}}(H)$, invented by Alon, Frankl and Lov\'{a}sz. We verify that $\text{Hom}(K_r^r,H)$ and $\mathsf{B}_{\text{edge}}(H)$, both are equipped with right actions of the symmetric group on $r$ letters $S_r$, are of the same simple $S_r$-homotopy type.