The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].

中文翻译：

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分离尺寸和度数

这分离维度图的G是最小正整数d其中嵌入了G进入ℝd，这样每对不相交的边都被某个轴平行的超平面隔开。我们证明了 Alon 等人的猜想。[SIAM J. 离散数学。2015] 通过显示每个具有最大程度 Δ 的图的分离维度小于 20Δ，这在一个常数因子下是最好的。我们还证明了分离维数为 3 的图具有有界平均度和有界色数，部分解决了 Alon 等人的开放问题。[J. 图论2018]。