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：

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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。