On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs

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 ${\psi }_{m,n}$, and deduce that the complex is homotopy equivalent to a wedge of ${\psi }_{m,n}$ spheres of dimension $(2m+2n+2mn-4)$, where $${\psi }_{m,n}=\left(\genfrac{}{}{0px}{}{2m+2n+2mn-1}{2}\right)-[(6m+2)n+(2m-4)]\mathrm{.}$$ 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.