The Stanley-Reisner ideal of the rook complex of polyominoes

TL;DR

This paper investigates the algebraic and combinatorial properties of rook complexes derived from polyominoes, focusing on their Stanley-Reisner ideals, purity, linear resolutions, and regularity in relation to graph invariants.

Contribution

It characterizes polyominoes with pure rook complexes and those with Stanley-Reisner ideals having linear resolutions, linking regularity to graph matching numbers.

Findings

Characterization of polyominoes with pure rook complexes

Identification of conditions for Stanley-Reisner ideals to have linear resolutions

Proof that regularity equals induced matching number for certain polyominoes

Abstract

We study the properties of the rook complex $\mathcal{R}$ of a polyomino $\mathcal{P}$ seen as independence complex of a graph $G$, and the associated Stanley--Reisner ideal $I_\mathcal{R}$. In particular, we characterize the polyominoes $\mathcal{P}$ having a pure rook complex, and the ones whose Stanley--Reisner ideal has linear resolution. Furthermore, we prove that for a class of polyominoes the Castelnuovo-Mumford regularity of $I_\mathcal{R}$ coincides with the induced matching number of $G$.