两个最小生成树可以交叉多少次？
How many times can two minimum spanning trees cross?
AI总结：
研究了双色点集的最小生成树交叉次数，证明了在通用点集情况下交叉次数的线性上界，并在密集或凸位置时给出下界，同时证明了随机点集下的期望线性性质。
摘要：
设P是平面上的一组n个点，且P=R∪B被染成两种颜色。我们关注R和B的最小生成树（MST）之间的交叉次数，记为crossAB(R,B)。我们将P的双色MST交叉数定义为cross(P)，记为cross(P)=max_{P=R∪B}(crossAB(R,B))。我们证明当P为通用点集时，cross(P)有一个线性上界。如果P密集或在凸位置，则提供线性下界。最后，如果P从单位正方形中均匀随机选择并均匀随机着色，则证明crossAB(R,B)的期望值是线性的。
英文摘要：
Let $P$ be a generic set of $n$ points in the plane, and let $P=R\cup B$ be a coloring of $P$ in two colors. We are interested in the number of crossings between the minimum spanning trees (MSTs) of $R$ and $B$, denoted by $\crossAB(R,B)$. We define the \emph{bicolored MST crossing number} of $P$, denoted by $\cross(P)$, as $\cross(P) = \max_{P= R\cup B}(\crossAB(R,B))$. We prove a linear upper bound for $\cross(P)$ when $P$ is generic. If $P$ is dense or in convex position, we provide linear lower bounds. Lastly, if $P$ is chosen uniformly at random from the unit square and is colored uniformly at random, we prove that the expected value of $\crossAB(R,B)$ is linear.
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。
