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arXiv 1602.04995cs.CGcs.DS
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关于非简单3-平面图的密度

On the Density of non-Simple 3-Planar Graphs

Michael A. Bekos, Michael Kaufmann, Chrysanthi N. Raftopoulou

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摘要：

\emph{$k$-planar graph}是可以在平面中绘制的图形，使得每条边最多交叉$k$次。对于$k \leq 4$，Pach和Tóth证明了$(k+3)(n-2)$关于$k$ -平面图的边总数的一个界，该界对于$k=1,2$是紧的.对于$k=3$，$6n-12$的界已被改进为$\frac{11}{2}n-11$，并已被证明是最佳的简单图的添加剂常数。在本文中，我们证明了$\frac{11}{2}n-11$边的界也适用于非简单的$3$ -平面图，其中允许非同伦平行边和自环。基于这个结果，\emph{optimal $3$-planar graphs}（即，具有$n$顶点和恰好$\frac{11}{2}n-11$边的$3$ -平面图）的特征可能是可能的，因为据我们所知，已知的简单$3$ -平面图不是最优的。

英文摘要：

A \emph{$k$-planar graph} is a graph that can be drawn in the plane such that every edge is crossed at most $k$ times. For $k \leq 4$, Pach and Tóth proved a bound of $(k+3)(n-2)$ on the total number of edges of a $k$-planar graph, which is tight for $k=1,2$. For $k=3$, the bound of $6n-12$ has been improved to $\frac{11}{2}n-11$ and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of $\frac{11}{2}n-11$ edges also holds for non-simple $3$-planar graphs that admit drawings in which non-homotopic parallel edges and self-loops are allowed. Based on this result, a characterization of \emph{optimal $3$-planar graphs} (that is, $3$-planar graphs with $n$ vertices and exactly $\frac{11}{2}n-11$ edges) might be possible, as to the best of our knowledge the densest known simple $3$-planar is not known to be optimal.

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。

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