标题： 关于受限万寿菊图的边数 显示英文标题

标题： On the number of edges of restricted matchstick graphs

摘要： 一个顶点是平面上的点且边是非相交单位长度直线段的图被称为\emph{火柴棒图}图。 我们证明了关于这种图在两种不同情形下最大边数的两个有些反直觉的结果。 首先，我们证明存在一个常数$c>0$，使得每个具有$n$个顶点的三角形自由匹配图最多有$2n-c\sqrt{n}$条边。 这个陈述对于任何$c>\sqrt2.$都不成立。 我们还证明了，对于每一个$r>0$，存在一个常数$\varepsilon(r)>0$，具有以下性质：包含在半径为$r$的圆盘中的$n$个顶点的每一个匹配图最多有$(2-\varepsilon(r))n$条边。 显示英文摘要

摘要： A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges of such graphs in two different scenarios. First, we show that there is a constant $c>0$ such that every triangle-free matchstick graph on $n$ vertices has at most $2n-c\sqrt{n}$ edges. This statement is not true for any $c>\sqrt2.$ We also prove that for every $r>0$, there is a constant $\varepsilon(r)>0$ with the property that every matchstick graph on $n$ vertices contained in a disk of radius $r$ has at most $(2-\varepsilon(r))n$ edges.