We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs $H$ of a graph $G$ of the neighborhood set system of $H$ is sandwiched between $\Omega(\log\mathrm{deg}(G))$ and $\mathcal{O}(\mathrm{deg}(G))$, where $\mathrm{deg}(G)$ denotes the degeneracy of $G$. We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization of bounded expansion classes. Then, we switch to a model theoretical point of view, introduce pointer structures, and study their relations to graph classes with bounded expansion. We deduce that a monotone class of graphs has bounded expansion if and only if all the set systems definable in this class have bounded hereditary discrepancy. Using known bounds on the VC-density of set systems definable in nowhere dense classes we also give a characterization of nowhere dense classes in terms of discrepancy. As consequences of our results, we obtain a corollary on the discrepancy of neighborhood set systems of edge colored graphs, a polynomial-time algorithm to compute $\varepsilon$-approximations of size $\mathcal{O}(1/\varepsilon)$ for set systems definable in bounded expansion classes, an application to clique coloring, and even the non-existence of a quantifier elimination scheme for nowhere dense classes.
翻译:我们研究组合差异概念与图形变异性概念之间的联系。 特别是, 我们证明所有子子组的最大差异是 $Omega( log\ mathrm{ deg} (G) $) 和 $mathcal{O}( mathrm{ deg} (G) 美元) 和 $mtercal{ deg} (G) 美元, 其中$\ mathrm{ deg} (G) 美元表示 $G$ 的变异性。 我们将这一结果扩大到与淡化的颜色数量和图形能力差异有关的颜色差异, 并推导出对闭合的扩展类的新的定性。 然后, 我们切换到一个模型理论点, 引入指针结构, 研究它们与带宽扩展的图表类之间的关系。 我们推论, 单调的图类, 只有当该类中所有可分解的系统都具有固定的遗传差异性时, 使用已知的VC- 定的颜色值值值等级, 直径直径的平面等级等级等级的平面值等级, 也给我们的货币系统的直径直径直径的直径直径直径直径判。