Topology of total cut complexes and cut complexes of grid graphs

TL;DR

This paper studies the topological properties of total cut and cut complexes specifically for grid graphs, extending previous work and confirming several conjectures to deepen understanding of these complexes.

Contribution

It provides new proofs and refinements of conjectures regarding the topology of cut complexes for 2x n and 3x n grid graphs, advancing the theoretical understanding.

Findings

Proved and strengthened conjectures on the topology of cut complexes for grid graphs.

Extended the analysis of total cut complexes to specific grid graph cases.

Enhanced the theoretical framework for understanding graph complexes in combinatorics.

Abstract

Inspired by the work of Fr{\"o}berg (1990) and Eagon and Reiner (1998), Bayer et al. recently introduced two new graph complexes: total cut complexes and cut complexes. In this article, we investigate these complexes specifically for (rectangular) grid graphs, focusing on $2 \times n$ and $3 \times n$ cases. We extend and refine the work of Bayer et al., proving and strengthening several of their conjectures, thereby enhancing the understanding of these graph complexes' topological and combinatorial properties.