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Weak Coloring Numbers of Intersection Graphs
arXiv - CS - Computational Geometry Pub Date : 2021-03-31 , DOI: arxiv-2103.17094 Zdeněk Dvořák, Jakub Pekárek, Torsten Ueckerdt, Yelena Yuditsky
arXiv - CS - Computational Geometry Pub Date : 2021-03-31 , DOI: arxiv-2103.17094 Zdeněk Dvořák, Jakub Pekárek, Torsten Ueckerdt, Yelena Yuditsky
Weak and strong coloring numbers are generalizations of the degeneracy of a
graph, where for each natural number $k$, we seek a vertex ordering such every
vertex can (weakly respectively strongly) reach in $k$ steps only few vertices
with lower index in the ordering. Both notions capture the sparsity of a graph
or a graph class, and have interesting applications in the structural and
algorithmic graph theory. Recently, the first author together with McCarty and
Norin observed a natural volume-based upper bound for the strong coloring
numbers of intersection graphs of well-behaved objects in $\mathbb{R}^d$, such
as homothets of a centrally symmetric compact convex object, or comparable
axis-aligned boxes. In this paper, we prove upper and lower bounds for the $k$-th weak coloring
numbers of these classes of intersection graphs. As a consequence, we describe
a natural graph class whose strong coloring numbers are polynomial in $k$, but
the weak coloring numbers are exponential. We also observe a surprising
difference in terms of the dependence of the weak coloring numbers on the
dimension between touching graphs of balls (single-exponential) and hypercubes
(double-exponential).
中文翻译:
交叉图的弱着色数
弱色数和强色数是图退化的一般化,其中对于每个自然数$ k $,我们寻求一个顶点排序,这样每个顶点可以(弱分别强)达到$ k $步长,而只有少数具有较低索引的顶点订购。这两个概念都捕获了图或图类的稀疏性,并且在结构图和算法图论中具有有趣的应用。最近,第一作者与McCarty和Norin一起观察了自然的基于体积的上限,该自然上限是$ \ mathbb {R} ^ d $中行为良好的对象的相交图的强着色数,例如中心对称紧凑型的均值凸物体或类似的轴对齐框。在本文中,我们证明了这些相交图类的第k个弱着色数的上限和下限。作为结果,我们描述了一个自然图类,其强色数是$ k $的多项式,而弱色数是指数的。我们还观察到在球(单指数)和超立方体(双指数)的触摸图之间弱色数对尺寸的依存关系方面出乎意料的差异。
更新日期:2021-04-01
中文翻译:
交叉图的弱着色数
弱色数和强色数是图退化的一般化,其中对于每个自然数$ k $,我们寻求一个顶点排序,这样每个顶点可以(弱分别强)达到$ k $步长,而只有少数具有较低索引的顶点订购。这两个概念都捕获了图或图类的稀疏性,并且在结构图和算法图论中具有有趣的应用。最近,第一作者与McCarty和Norin一起观察了自然的基于体积的上限,该自然上限是$ \ mathbb {R} ^ d $中行为良好的对象的相交图的强着色数,例如中心对称紧凑型的均值凸物体或类似的轴对齐框。在本文中,我们证明了这些相交图类的第k个弱着色数的上限和下限。作为结果,我们描述了一个自然图类,其强色数是$ k $的多项式,而弱色数是指数的。我们还观察到在球(单指数)和超立方体(双指数)的触摸图之间弱色数对尺寸的依存关系方面出乎意料的差异。