On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs

Abstract

The $k$-cut complex was recently introduced by Bayer et al. as a generalization of earlier work of Fr{\"o}berg (1990) and Eagon and Reiner (1998), and was shown to be shellable for several classes of graphs. In this article, we prove that the $3$-cut complexes of the hexagonal grid graphs $H_{1 \times m \times n}$ are shellable for all $m,n \geq 1$, by constructing an explicit shelling order using reverse lexicographic ordering. From this shelling, we determine the number of spanning facets, denoted by $\psi_{m,n}$, and deduce that the complex is homotopy equivalent to a wedge of $\psi_{m,n}$ spheres of dimension $\left( 2m + 2n + 2mn - 4 \right)$, where $$\psi_{m,n} = \binom{2m+2n+2mn-1}{2} - \left[ \left( 6m+2 \right) n + (2m-4) \right].$$ While these topological properties can be obtained from general results of Bayer et al., we provide an explicit combinatorial construction of a shelling order, yielding a direct counting formula for the number of spheres in the wedge sum decomposition.