论文标题
将完整图的图纸扩展到伪圆的排列
Extending drawings of complete graphs into arrangements of pseudocircles
论文作者
论文摘要
由于几何形状成功地证明了$ k_n $的“伪线”图纸的刺激性猜想的动机，我们介绍了图形的“伪圈”图纸。图形$ g $的球形图是单位球中的图形$ \ mathbb {s}^2 $，其中$ g $的顶点表示为点 - $ g $的边缘是$ g $的边缘，而$ g $的边缘是$ \ mathbb {s}^2 $连接的Vertices的$ \ Mathbb {s}^2 $ connect。这样的图纸具有三个属性：（1）每个边缘$ e $都包含在简单的封闭曲线$γ_e$中，因此$γ_e$中唯一的顶点是$ e $的末端； （2）如果$ e \ ne f $，则$γ_e\capγ_f$完全具有两个交叉点； （3）如果$ e \ ne f $，则$ e $最多一次在交叉处或$ e $的结尾。我们使用属性（1） - （3）来定义$ g $的伪圈图。我们的主要结果是，对于完整的图，属性（1） - （3）等于相同的三个属性，但在（2）中以“最多两个交叉”代替了“精确的两个交叉”。 证明需要将伪圆形排列的几何横向理论产生结果。证明了这一点使用令人惊讶的结果，即在简单闭合曲线的布置中缺少特殊弧线（相干螺旋），这表征了以下事实：排列中的任何两个曲线最多都有两个交叉点。 我们的研究为展示$ k_ {10} $的图纸提供了必要的想法，该图形没有扩展到伪圆的布置和$ k_9 $的图纸，这确实扩展到了伪圆的布置，但是没有这样的扩展使所有成对的伪圆形交叉交叉。
Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for "pseudolinear" drawings of $K_n$, we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\mathbb{S}^2$ in which the vertices of $G$ are represented as points -- no three on a great circle -- and the edges of $G$ are shortest-arcs in $\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $γ_e$ such that the only vertices in $γ_e$ are the ends of $e$; (2) if $e\ne f$, then $γ_e\capγ_f$ has precisely two crossings; and (3) if $e\ne f$, then $e$ intersects $γ_f$ at most once, either in a crossing or an end of $e$. We use Properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that, for the complete graph, Properties (1)--(3) are equivalent to the same three properties but with "precisely two crossings" in (2) replaced by "at most two crossings". The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs ( coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.
