arXiv 2512.044862025-12-04math.CO

总2-切割复形的循环图幂和某些图的笛卡尔积

Total $2$-cut complexes of powers of cycle graphs and Cartesian products of certain graphs

IIT Mandi（印度理工学院曼迪分校）；University of Kentucky（肯塔基大学）

Pratiksha Chauhan, Samir Shukla, Kumar Vinayak

备注：: 16 pages

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AI总结：

本文研究了循环图幂和某些图的笛卡尔积的总2-切割复形的同调类型，利用离散Morse理论证明其与球面的楔和同构，并验证了关于循环图幂的猜想。

摘要：

对于正整数k，图G的总k-切割复形Δ_k^t(G)是其一个单纯复形，其面是σ⊆V(G)满足|σ|=|V(G)|-k且诱导子图G[V(G)\σ]不包含任何边。这些复形由Bayer等人在文献[1]中引入，与交换代数相关。在同一篇论文中，他们研究了这些复形对于各种图类的同调类型，包括循环图C_n、平方循环图C_n^2以及完全图和路径图的笛卡尔积K_m□P_2和K_2□P_n。在本文中，我们扩展了Bayer等人的工作，针对这些图类。我们专注于复形Δ_2^t(G)，并确定了这三类图的同调类型：(i) 循环图的p次幂C_n^p；(ii) K_m□P_n；(iii) K_m□C_n。利用离散Morse理论，我们证明这些复形与球面的楔和同调等价。我们还给出了同调类型中出现的球面数量和维度。我们关于循环图幂C_n^p的结果证明了Shen等人关于复形Δ_2^t(C_n^p)同调类型的猜想。

英文摘要：

For a positive integer $k$, the \emph{ total $k$-cut complex} of a graph $G$, denoted as $Δ_k^t(G)$, is the simplicial complex whose facets are $σ\subseteq V(G)$ such that $|σ| = |V(G)|-k$ and the induced subgraph $G[V(G) \setminus σ]$ does not contain any edge. These complexes were introduced by Bayer et al.\ in \cite{Bayer2024TotalCutcomplex} in connection with commutative algebra. In the same paper, they studied the homotopy types of these complexes for various families of graphs, including cycle graphs $C_n$, squared cycle graphs $C_n^2$, and Cartesian products of complete graphs and path graphs $K_m \square P_2$ and $K_2 \square P_n$. In this article, we extend the work of Bayer et al.\ for these families of graphs. We focus on the complexes $Δ_2^t(G)$ and determine the homotopy types of these complexes for three classes of graphs: (i) $p$-th powers of cycle graphs $C_n^p$ (ii) $K_m \square P_n$ and (iii) $K_m \square C_n$. Using discrete Morse theory, we show that these complexes are homotopy equivalent to wedges of spheres. We also give the number and dimension of spheres appearing in the homotopy type. Our result on powers of cycle graphs $C_n^p$ proves a conjecture of Shen et al.\ about the homotopy type of the complexes $Δ_2^t(C_n^p)$.

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