论文标题
界定简单复合物和图形的可折叠性数量
Bounding the collapsibility number of simplicial complexes and graphs
论文作者
论文摘要
我们介绍并研究了一种新的组合不变的简单络合物的theta-number $θ（x）$，并证明不平等$ \ MATHCAL {c}（x）\leqθ（x）$ commistial complose $ x $ complose complops $ x $ complaps $ $ nordialigience我们显示了与theta-number合作的优势。它的纯粹组合配方使我们能够验证现有界限对LERAY和可折叠性数字的有效性，并提供涉及其他参数的新界限。 我们表明，顶点可分解的简单复合物的theta-number，可折叠性和leray数都是相等的。此外，我们证明了图$ g $的独立综合体的theta-number与其诱导的匹配数字$ im（g）$密切相关，因为它发生在此类复合物的leray数字上。我们确定它们相等的图形类，否则提供涉及上限的上限。特别是，我们证明，对于每个$ n $ n $ -vertex Graph $ g $，在上面的theta-number在上面的限制为$ 2 \ sqrt {n \ cdot im（g）} $，在$ 2K_2 $ -FREE-free Graphs的情况下，我们将其降低到$ 2 \ log nog n $。此外，我们验证了theta-number是基础图上的收缩小单调。
We introduce and study a new combinatorial invariant the theta-number $θ(X)$ of simplicial complexes, and prove that the inequality $\mathcal{C}(X)\leq θ(X)$ holds for every simplicial complex $X$, where $\mathcal{C}(X)$ denotes the collapsibility number of $X$. We display the advantages of working with the theta-number. Its purely combinatorial formulation enables us to verify the validity of the existing bounds on both Leray and collapsibility numbers as well as provide new bounds involving other parameters. We show that the theta-number, collapsibility and Leray numbers of a vertex decomposable simplicial complex are all equal. Moreover, we prove that the theta-number of the independence complex of a graph $G$ is closely related to its induced matching number $im(G)$ as it happens to the Leray number of such complexes. We identify graph classes where they are equal, and otherwise provide upper bounds involving it. In particular, we prove that the theta-number is bounded from above by $2\sqrt{n\cdot im(G)}$ for every $n$-vertex graph $G$, and in the case of $2K_2$-free graphs, we lower this bound to $2\log n$. Furthermore, we verify that the theta-number is contraction minor monotone on the underlying graph.
