Total cut complexes and their duals

TL;DR

This paper investigates the homotopy types of total cut complexes and their Alexander duals for various graph classes, solving multiple conjectures and providing new insights into their topological properties.

Contribution

It computes the homotopy types of total cut complexes for powers of cycles, complete multipartite graphs, and Cartesian products, addressing several open conjectures.

Findings

Homotopy type of total cut complexes for certain powers of cycles is determined.

Homotopy type of the 2-total cut complex for rth powers of cycles with r≥3 is calculated.

Results on connectivity and homotopy types for complexes of complete multipartite graphs and Cartesian products.

Abstract

We study the total cut complexes and their Alexander duals. The homotopy type of these complexes is calculated for de $p$th power of a cycle with at least $2rn$ vertices where $p\leq r$, solving part of a conjecture of Bayer, Denker, Milutinovi\'c, Rowlands, Sundaram and Xue. The homotopy type of the $2$-total cut complex for any $r$th power of a cycle with $r\geq3$ also is calculated, solving a conjecture of Chauhan, Shukla and Vinayak. We give some results about the connectivity. The homotopy type of the complexes for complete multipartite graph is determined. We also study the complexes of cartesian products of paths and of cartesian products of complete graphs for the total $2$-cut complex.