Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph $\operatorname{KG}(n,2)$ that have a chromatic number equal to its chromatic number, namely $n-2$. In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the $\operatorname{NP}$-hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this paper are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results.

中文翻译：

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线图的补色着色

我们的目的是表明线图的补集具有很好的着色特性。我们表明，对于此类中的所有图，局部色数和通常色数是相等的。我们还证明了色数等于自然上限的充分条件。后一种条件的结果是 Kneser 图 $\operatorname{KG}(n,2)$ 的所有诱导子图的完整表征，这些子图的色数等于其色数，即 $n-2$。除了上界之外，Dol'nikov 定理还提供了下界，这是图论中拓扑方法的经典结果。我们证明了 $\operatorname{NP}$-hardness 决定色数和任何这些边界之间的相等性。拓扑方法特别适用于研究超图的线图的补的着色性质。尽管如此，本文中的所有证明都是基本的，我们还对拓扑方法覆盖我们的一些结果的能力进行了简短的讨论。