论文标题
在没有空镜头的$ k_n $的星空图纸中的最大交叉数
On the Maximum Number of Crossings in Star-Simple Drawings of $K_n$ with No Empty Lens
论文作者
论文摘要
图形的星光图是相邻边缘不交叉的图形。相反，两个独立边缘之间的交叉数没有限制。当允许空透镜（由两个边缘由2个周界界定的边缘引起的表面）时，两个独立的边缘可能在星光般的图纸中任意多次交叉。我们考虑没有空镜头的$ k_n $的星光般的图纸。在这种情况下，我们证明了$ 3（（n-4）！）$的上限在任何一对边缘之间的最大交叉数。因此，交叉的总数是有限的，上限为$ n！$。
A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of $K_n$ with no empty lens. In this setting we prove an upper bound of $3((n-4)!)$ on the maximum number of crossings between any pair of edges. It follows that the total number of crossings is finite and upper bounded by $n!$.
