On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915774959976448 |
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| author | Chandrakar, Himanshu |
| author_facet | Chandrakar, Himanshu |
| contents | 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 $m,n \geq 1$, by constructing an explicit shelling order using reverse lexicographic ordering. From this shelling, we determine the number of spanning facets, denoted by $ψ_{m,n}$, and deduce that the complex is homotopy equivalent to a wedge of $ψ_{m,n}$ spheres of dimension $\left( 2m + 2n + 2mn - 4 \right)$, where $$ψ_{m,n} = \binom{2m+2n+2mn-1}{2} - \left[ \left( 6m+2 \right) n + (2m-4) \right].$$
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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21755 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs Chandrakar, Himanshu Combinatorics 05C69, 05E45, 55P15, 57M15 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 $m,n \geq 1$, by constructing an explicit shelling order using reverse lexicographic ordering. From this shelling, we determine the number of spanning facets, denoted by $ψ_{m,n}$, and deduce that the complex is homotopy equivalent to a wedge of $ψ_{m,n}$ spheres of dimension $\left( 2m + 2n + 2mn - 4 \right)$, where $$ψ_{m,n} = \binom{2m+2n+2mn-1}{2} - \left[ \left( 6m+2 \right) n + (2m-4) \right].$$ 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. |
| title | On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs |
| topic | Combinatorics 05C69, 05E45, 55P15, 57M15 |
| url | https://arxiv.org/abs/2512.21755 |