A perfect K_r‐tiling in a graph G is a collection of vertex‐disjoint copies of K_r that together cover all the vertices in G. In this paper we consider perfect K_r‐tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed we determine how many random edges one must add to an n‐vertex graph G of minimum degree to ensure that, asymptotically almost surely, the resulting graph contains a perfect K_r‐tiling. As one increases we demonstrate that the number of random edges required “jumps” at regular intervals, and within these intervals our result is best‐possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu [25] (which resolves the purely random case, that is, ) and that of Hajnal and Szemerédi [18] (which demonstrates that for the initial graph already houses the desired perfect K_r‐tiling).

中文翻译：

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随机扰动图中的平铺：弥合Hajnal-Szemerédi和Johansson-Kahn-Vu之间的差距

一个完美的ķ _[R在图-tiling摹是顶点不相交的副本的集合ķ _[R一起覆盖所有顶点摹。在本文中，我们在随机扰动图的设置中考虑了完美的K _r-平铺；由Bohman，Frieze和Martin [7]引入的模型，其中一个模型从一个密集图开始，然后向其添加m个随机边。具体而言，给定任何固定值，我们确定一个边数必须添加到最小度的n个顶点图G上，以确保渐近几乎确定地，所得图包含完美的K _r-平铺。随着增加，我们证明随机边缘的数量需要以固定间隔“跳跃”，并且在这些间隔内，我们的结果是最好的。因此，这项工作弥合了Johansson，Kahn和Vu [25]的开创性工作（它解决了纯随机的情况，即）与Hajnal和Szemerédi[18]的开创性工作之间的差距（这表明初始图已经存在期望完美ķ _ř -tiling）。