Homotopy types of Hom complexes of graphs

TL;DR

This paper investigates the homotopy types of Hom complexes of graphs, revealing limitations of homotopy invariants in bounding graph chromatic numbers and exploring equivariant homotopy types.

Contribution

It demonstrates that no homotopy invariant of Hom complexes can upper bound the chromatic number, and constructs examples showing homotopy equivalence does not imply similar chromatic bounds.

Findings

Hom complexes' homotopy invariants do not bound chromatic number from above.

Existence of homotopy equivalent Hom complexes with significantly different chromatic numbers.

Analysis of equivariant homotopy types of Hom complexes.

Abstract

The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a pair of graphs $T$ and $G$, considered in the graph coloring problem. It is known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds for the chromatic number of $G$. For a fixed finite graph $T$, we show that there is no homotopy invariant of ${\rm Hom}(T,G)$ which gives an upper bound for the chromatic number of $G$. More precisely, for a non-bipartite graph $G$, we construct a graph $H$ such that ${\rm Hom}(T,G)$ and ${\rm Hom}(T,H)$ are homotopy equivalent but $\chi(H)$ is much larger than $\chi(G)$. The equivariant homotopy type of ${\rm Hom}(T,G)$ is also considered.