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).

翻译：微弱和强烈的彩色数字是图表变色的概括性。 对于每个自然数字$k$, 我们寻找一个顶点, 以美元为单位, 每个顶点都能( 各自变强) 以美元为单位, 只需要几个有较低指数的脊椎。 两种概念都捕捉图或图类的宽度, 并在结构和算法图形理论中具有有趣的应用。 最近, 第一作者与麦卡蒂和诺林 一起观察到一个自然的以体积为单位, 以美元为单位, 以美元为单位, 以美元为单位, 且颜色数字较弱, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以自然的体积为单位, 以自然的体积为上限, 以美元为单位, 以美元为单位, 以色度较弱的体积数字以指数为指数。 我们还观察了中央对中央相对等缩缩缩缩缩缩缩缩的色度, 以平面的比值表示。