A {\em thrackle} is a graph drawn in the plane so that every pair of itsedges meet exactly once: either at a common end vertex or in a proper crossing.We prove that any thrackle of $n$ vertices has at most $1.3984n$ edges. {\emQuasi-thrackles} are defined similarly, except that every pair of edges that donot share a vertex are allowed to cross an {\em odd} number of times. It isalso shown that the maximum number of edges of a quasi-thrackle on $n$ verticesis ${3\over 2}(n-1)$, and that this bound is best possible for infinitely manyvalues of $n$.
展开▼
机译：{\ em thrckle}是在平面上绘制的图形，因此，每对边缘均恰好相交一次：在公共末端顶点或在适当的交叉点。我们证明，$ n $个顶点的任何转向缝最多具有$ 1.3984n $边。 {\ emQuasi-thrackles}的定义类似，不同之处在于，不共享顶点的每对边都可以穿越{\ em奇数}次。还显示了在$ n $顶点$ {3 \ over 2}（n-1）$上准搏动的最大边数，并且对于$ n $的无限多个值，该边界是最佳的。
展开▼
