AI 中文总结
本文确定了莫比乌斯梯子图 $M_{2n}$ 的独立复形与完美匹配复形的同伦型，前者按 $n$ 模4周期变化，偶数 $n$ 时后者为2个球面楔和，奇数 $n$ 时按 $n$ 模6周期变化。
AI 中文摘要
图的独立复形和完美匹配复形是分别编码其独立集和完美匹配的单纯复形。一般而言，确定这些复形的同伦型及其他拓扑性质是一个难题，仅对组合正则性或对称性较强的相对受限图类才有明确描述。尽管针对方格图等若干图族已对这些复形开展了大量研究，但对其他高度对称的自然图族的了解相对较少。本文确定了莫比乌斯梯子图 $M_{2n}$ 的独立复形与完美匹配复形的同伦型。莫比乌斯梯子图是从 $2n$ 阶环通过连接成对对顶点得到的自然高度对称的三次图族。我们证明，独立复形 $\text{Ind}(M_{2n})$ 的同伦型按 $n$ 模 4 呈周期性依赖：当 $n=4k$ 时，它同伦等价于 $\boldsymbol{S}^{2k-1}$；当 $n=4k+2$ 时，它同伦等价于 3 个 $\boldsymbol{S}^{2k}$ 的楔和；当 $n=4k+1$ 或 $4k+3$ 时，它同伦等价于 $\boldsymbol{S}^{2k}$。我们进一步确定了完美匹配复形 $\boldsymbol{\textit{M}}_p(M_{2n})$ 的同伦型：当 $n$ 为偶数时，它同伦等价于 2 个 $\boldsymbol{S}^{(n-2)/2}$ 的楔和；当 $n$ 为奇数时，其同伦型按 $n$ 模 6 呈周期性依赖。因此，我们的结果为重要的高度对称三次图族对应的两个基础单纯复形提供了明确描述。
英文摘要
The independence complex and perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining their homotopy types is generally difficult, with explicit descriptions known mainly for highly structured graph families. In this article, we determine the homotopy types of these complexes for the Möbius ladder graphs $M_{2n}$ and circular ladder graphs $\mathcal{C}_{2n}$. The Möbius ladder graphs $M_{2n}$ are highly symmetric cubic graphs obtained from a $2n$-cycle by joining opposite vertices, while the circular ladder graphs $\mathcal{C}_{2n}$ are the Cartesian products of an $n$-cycle and a path of length one. We show that $\operatorname{Ind}(M_{2n})$ and $\operatorname{Ind}(\mathcal{C}_{2n})$ have the homotopy type of wedges of spheres, with the numbers and dimensions of the spheres exhibiting periodic behavior according to $n$ modulo $4$. For the perfect matching complex $\mathcal{M}_p(M_{2n})$, its homotopy type is a wedge of two copies of $\mathbb{S}^{(n-2)/2}$ when $n$ is even, while for odd $n$ it has the homotopy type of a wedge of spheres whose numbers and dimensions depend periodically on $n$ modulo $6$. The perfect matching complex $\mathcal{M}_p(\mathcal{C}_{2n})$ is contractible for odd $n$, whereas for even $n$ its homotopy type is a wedge of spheres, with the numbers and dimensions determined periodically by $n$ modulo $6$. Thus, we obtain explicit homotopy types for the independence and perfect matching complexes of two highly symmetric families of cubic graphs, which are also relevant in crystallization theory and the combinatorial representation of PL manifolds.
Comments19 pages, 6 Figures
