发表机构
Leipzig University; University of Warwick（莱比锡大学; 华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Chepoi和Hagen的问题，构造了两个有界度CAT(0)方形复形，证明其不能等距嵌入树的有限乘积，并揭示Burling图是唯一障碍，同时构造了高围长高色数的弦图与区间图交集。
AI 中文摘要
回答Chepoi和Hagen提出的一个问题，我们给出了两个有界度CAT(0)方形复形的构造，它们不能等距嵌入到任何树的有限乘积中。第一个构造基于Burling图，其度至多为5，这是最优的。这些方形复形拓扑嵌入到$\R^3$中，因为我们进一步证明了每个顶点度至多为5的CAT(0)方形复形都可以拓扑嵌入到一条直线与一个星形的乘积中。事实证明，在这种情况下，Burling图是唯一的障碍：我们证明了拓扑嵌入到$\R^3$中且其交叉图禁止某个诱导Burling图的CAT(0)方形复形可以等距嵌入到树的有限乘积中。我们的第二个CAT(0)方形复形度至多为6，且其交叉图禁止诱导Burling图。具有独立的图论意义的是，在此过程中我们构造了弦图与区间图的交集，这些交集具有任意大的围长和色数。我们还讨论了与事件结构的良好标号之间的联系。
英文摘要
Answering a question of Chepoi and Hagen, we give two constructions of bounded degree CAT(0) square complexes that cannot be isometrically embedded into any finite product of trees. The first is based on Burling graphs and has degree at most five, which is optimal. These square complexes topologically embed into $\R^3$ since we further prove that every CAT(0) square complex whose vertices have degree at most five can be topologically embedded into the product of a line and a star. It turns out that in this setting, Burling graphs are the only obstruction: we prove that CAT(0) square complexes that topologically embed into $\R^3$ and whose crossing graph forbids some induced Burling graph can be isometrically embedded into a finite product of trees. Our second CAT(0) square complex has degree at most six and its crossing graph forbids an induced Burling graph. Of independent graph-theoretic interest, along the way we construct intersections of chordal and interval graphs with arbitrarily large girth and chromatic number. We also discuss connections to nice labellings of event structures.
Comments25 pages
