arXiv 1704.018922017-04math.CO

线段的不相交图

Disjointness graphs of segments

Janos Pach, Gabor Tardos, Geza Toth

摘要：

$R^d$中的线段集${\cal S}$的{ \em不交图} $G=G({\cal S})$，$d\ge 2,$是顶点集为${\cal S}$且两个顶点由一条边连通的图当且仅当对应的线段不交.证明了$G$的色数满足$χ(G)\le(ω(G))^4+(ω(G))^3$，其中$ω(G)$表示$G$的团数.因此，$\cal S$具有$Ω(n^{1/5})$成对相交或成对不相交的元素。更强的界限是针对空间中的直线而不是线段建立的。我们表明，计算$ω(G)$和$χ(G)$的不相交图在空间中的线是NP-困难的任务。然而，我们可以设计有效的算法来计算$G$的适当着色，其中颜色的数量满足上述上界。即使在平面中，也不能期望连续弧集而不是线段集得到类似的结果。我们构建家庭的弧的不相交图是三角形自由（$ω(G)=2$），但其色数是任意大的。

英文摘要：

The {\em disjointness graph} $G=G({\cal S})$ of a set of segments ${\cal S}$ in $R^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of $G$ satisfies $χ(G)\le(ω(G))^4+(ω(G))^3$, where $ω(G)$ denotes the clique number of $G$. It follows, that $\cal S$ has $Ω(n^{1/5})$ pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing $ω(G)$ and $χ(G)$ for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of $G$ in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free ($ω(G)=2$), but whose chromatic numbers are arbitrarily large.

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