分离维度和程度
Separation Dimension and Degree
摘要：
图$G$的“分离维数”是最小正整数$d$，对于该最小正整数$d$，存在$G$嵌入到$\mathbb{R}^d$中，使得每对不相交的边被某个轴平行的超平面分离。我们证明了Alon等人的猜想。[SIAM J. Discrete Math. 2015]通过证明每个具有最大度$Δ$的图的分离维数小于$20Δ$，这是最好的可能常数因子。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等人的一个公开问题。[J.图论2018]。
英文摘要：
The "separation dimension" of a graph $G$ is the minimum positive integer $d$ for which there is an embedding of $G$ into $\mathbb{R}^d$, 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].
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