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{ö}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 $m,n\ge 1$, by constructing an explicit shelling order using reverse lexicographic ordering. From this shelling, we determine the number of spanning facets, denoted by $\ddot{I}{ˆ}_{m,n}$, and deduce that the complex is homotopy equivalent to a wedge of $\ddot{I}{ˆ}_{m,n}$ spheres of dimension $\left(2m+2n+2mn-4\right)$, where $$\ddot{I}{ˆ}_{m,n}=\left(\genfrac{}{}{0ex}{}{2m+2n+2mn-1}{2}\right)-\left[\left(6m+2\right)n+(2m-4)\right].$$ 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.

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