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Tilings in graphons
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-12-11 , DOI: 10.1016/j.ejc.2020.103284
Jan Hladký , Ping Hu , Diana Piguet

We introduce a counterpart to the notion of tilings, that is vertex-disjoint copies of a fixed graph F, to the setting of graphons. The case F=K2 gives the notion of matchings in graphons. We give a transference statement that allows us to switch between the finite and limit notion, and derive several favorable properties, including the LP-duality counterpart to the classical relation between the fractional vertex covers and fractional matchings/tilings, and discuss connections with property testing.

As an application of our theory, we determine the asymptotically almost sure F-tiling number of inhomogeneous random graphs G(n,W). As another application, in an accompanying paper (Hladký et al., 2019) we give a proof of a strengthening of a theorem of Komlós (Komlós, 2000).



中文翻译:

石墨烯中的瓷砖

我们介绍了平铺的概念,即固定图的顶点不相交的副本 F,以设置Graphon。案子F=ķ2给出了石墨烯中匹配的概念。我们给出一个转移语句,使我们可以在有限和极限概念之间切换,并得出几个有利的属性,包括与分数顶点覆盖和分数匹配/平铺之间的经典关系对应的LP对偶,并讨论与属性测试的关系。

作为我们理论的应用,我们确定渐近几乎确定 F不均匀随机图的平铺数目 Gñw ^。作为另一项应用,在随附的论文中(Hladký等,2019),我们证明了Komlós定理的增强(Komlós,2000)。

更新日期:2020-12-13
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