论文标题
部分可观测时空混沌系统的无模型预测
Homotopy and the Homomorphism Threshold of Odd Cycles
论文作者
论文摘要
考虑一个$ c_ {2r+1} $的家庭$ \ Mathcal f $ - 免费图形，其中$ r \ geq 2 $。假设$ \ Mathcal F $中的每个图在其顶点数量中具有最低度线性。托马森（Thomassen）表明，这样的家族具有限制色数，或者等效地，$ \ Mathcal f $中的所有图都与有界尺寸的完整图相同。考虑到本身就是$ c_ {2r+1} $的同构图像 - 免费，我们构建了一个密集的$ c_ {2r+1} $的家族 - 没有$ C_ {2R+1} $的免费图形 - 有边界大小的免费同质图像。这为长度至少5个奇数周期的同态阈值提供了第一个非平凡的下限，并回答了Ebsen和Schacht的问题。 我们的证明引入了一种新技术来描述图形的拓扑结构。我们建立了同型等效性的图理论类似物，这使我们能够分析奇数封闭步行的相对位置。这个概念与邻里建筑群有意外的联系，导致了多个有趣的问题。
Consider a family $\mathcal F$ of $C_{2r+1}$-free graphs, where $r\geq 2$. Suppose that each graph in $\mathcal F$ has minimum degree linear in its number of vertices. Thomassen showed that such a family has bounded chromatic number, or, equivalently, that all graphs in $\mathcal F$ are homomorphic to a complete graph of bounded size. Considering instead homomorphic images which are themselves $C_{2r+1}$-free, we construct a family of dense $C_{2r+1}$-free graphs with no $C_{2r+1}$-free homomorphic image of bounded size. This provides the first nontrivial lower bound on the homomorphism threshold of odd cycles of length at least 5 and answers a question of Ebsen and Schacht. Our proof introduces a new technique to describe the topological structure of a graph. We establish a graph-theoretic analogue of homotopy equivalence, which allows us to analyze the relative placement of odd closed walks in a graph. This notion has unexpected connections to the neighborhood complex, leading to multiple interesting questions.
