Deformations of box complexes

TL;DR

This paper investigates the relationship between the $bZ_2$-homotopy invariants of box complexes associated with graphs and the graphs' chromatic numbers, revealing limitations of these invariants in capturing chromatic properties.

Contribution

It demonstrates that $bZ_2$-homotopy invariants of box complexes are not sufficient to determine the chromatic number, introducing new deformations that preserve homotopy types but alter chromatic numbers.

Findings

$bZ_2$-homotopy invariants do not fully determine chromatic number.

Constructed graphs with homotopy equivalent box complexes but different chromatic numbers.

Identified deformations that preserve $bZ_2$-simple homotopy types of box complexes.

Abstract

Box complex is a $\mathbb{Z}_2$-space associated to a graph, and it is known that a certain $\mathbb{Z}_2$-homotopy invariant of it, called the $\mathbb{Z}_2$-index, gives an effective lower bound for the chromatic number. On the other hand, we show that any $\mathbb{Z}_2$-homotopy invariant of the box complex is not equivalent to the chromatic number. Namely, we construct a graph homomorphism $f:X \rightarrow Y$ such that it gives rise to a $\mathbb{Z}_2$-homotopy equivalence between their box complexes, but $X$ and $Y$ have different chromatic numbers. To see this, we show that some deformations of graphs do not change the $\mathbb{Z}_2$-simple homotopy types of box complexes.