由几何图诱导的相交直径球
Intersecting diametral balls induced by a geometric graph
摘要：
对于一个顶点集是欧氏$d$ -空间中有限点集的图，考虑其边诱导直径的闭（开）球。如果这些闭（开）球相交，则该图称为（开）Tverberg图。利用二等分线的思想，证明了（$i$）对于平面上的任意有限点集，存在一个哈密顿圈即Tverberg图;（$ii$）对于平面上的任意$ n $红点和$ n $蓝点，存在一个完美红蓝匹配即Tverberg图.此外，我们证明了（$iii$）的任何偶数点集在欧几里德$ d $ -空间，存在一个完美的匹配，这是一个开放的Tverberg图;（$iv$）的任何$ n $红色和$ n $蓝色的点在欧几里德$ d $ -空间，存在一个完美的红蓝匹配，这是一个Tverberg图。
英文摘要：
For a graph whose vertex set is a finite set of points in the Euclidean $d$-space consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that ($i$) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; ($ii$) for any $ n $ red and $ n $ blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Also, we prove that ($iii$) for any even set of points in the Euclidean $ d $-space, there exists a perfect matching that is an open Tverberg graph; ($iv$) for any $ n $ red and $ n $ blue points in the Euclidean $ d $-space, there exists a perfect red-blue matching that is a Tverberg graph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
