Tverberg 图形上的图 (On a Tverberg graph)

from arxiv, v2: 11 pages, 3 figures. Key words: Tverberg's Theorem, perfect matching, blue-red matching, Hamiltonian cycle, alternating cycle, infinite descent, halving lines

For a graph whose vertex set is a finite set of points in $\mathbb R^d$, 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 $ \mathbb R^d $, there exists a perfect matching that is an open Tverberg graph; (iv) for any $ n $ red and $ n $ blue points in $ \mathbb R^d $, there exists a perfect red-blue matching that is a Tverberg graph.

翻译：对于其顶点设置为 $mathbb R ⁇ d$ 的有限点数的图形, 请考虑其边缘引导直径的封闭( 开放) 球球。如果这些闭( 开放) 球相互交错, 则该图称为( 开放) Tverberg 图形。我们使用将线减半的想法, 显示 (一) 对于飞机上的任何有限点数, 存在一个汉密尔顿周期, 即 Tverberg 图形 ;(二) 对于飞机上的任何 n 美元红色和 n 美元蓝点, 都存在完美的红蓝色匹配, 即 Tverberg 图形。另外, 我们还证明 (三) 对于任何以 $\ mathbbr R ⁇ d $ 的甚至一组点数, 都存在一个完美的匹配; (四) 对于任何 n 美元红和 n 美元蓝点, 以 $\ mathb R ⁇ d$ 的, 存在一个完美的红蓝色匹配, 即 Tverberg 图形。