The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$, $$d \ge 2$$, is a graph whose vertex set is 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 $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$, where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) 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 colourings of G in which the number of colours 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.

中文翻译：

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空间段的不相交图

这不相交图 G = G(𝒮) 一组段𝒮在${\mathbb{R}^d}$,$$d \ge 2$$, 是一个图，其顶点集是𝒮当且仅当对应的线段不相交时，两个顶点由一条边连接。我们证明了色数G满足$\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$， 在哪里ω(G) 表示的团数G. 它遵循𝒮有Ω(n1/5) 成对相交或成对不相交的元素。为空间中的线而不是线段建立了更强的界限。我们表明，计算ω(G） 和χ(G) 因为空间线的不相交图是 NP-hard 任务。但是，我们可以设计有效的算法来计算正确的着色G其中颜色的数量满足上述上限。即使在平面中，人们也不能期望对于连续弧而不是线段的集合会产生类似的结果。我们构建了不相交图是无三角形的弧族（ω(G) = 2)，但其色数任意大。