A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-tiling in a graph G. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks substructures F of G that have ‘high’ discrepancy (i.e. the sum of the labels of the edges in F is far from 0). In this paper we determine the minimum degree threshold for a graph to contain a perfect Kr-tiling of high discrepancy.

中文翻译：

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Hajnal–Szemerédi 定理的差异版本

一个完美的ķr-平铺在图表中G是 clique 的顶点不相交副本的集合ķr在G覆盖每个顶点G. 著名的 Hajnal-Szemerédi 定理确定了强制完美的最小度阈值ķr-平铺在图表中G. 差异的概念出现在许多数学分支中。在图形设置中，分配图形的边G来自 {‒1, 1} 的标签，一个寻找子结构F的G具有“高”差异的（IE中边的标签总和F远非 0）。在本文中，我们确定了图包含完美的最小度阈值ķr- 高差异的平铺。