Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles
Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles
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DOI：
10.1017/s096354831900021x
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发表时间：
2016-12
期刊：
影响因子：
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通讯作者：
Wei Gao;Jie Han;Yi Zhao
中科院分区：
文献类型：
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作者：
Wei Gao;Jie Han;Yi Zhao
给定两个k-图（k-一致超图）F和H，H中的完美F-平铺（或F-因子）是F的一组顶点不相交的副本，它们一起覆盖H的顶点集。对于所有的完全k部k-图K，Mycroft证明了一个最小余度条件，该条件保证了n-顶点k-图中的K-因子，并且紧到误差项o（n）。本文将Mycroft结果中的误差项改进为一个次线性项，当K的顶点类的大小差互质时，它与K的Turán数有关.此外，我们找到了一个结构，表明我们的改进的余度条件是渐近紧在无限多的情况下，从而反驳了Mycroft猜想。最后，我们确定了平铺K（k）（1，...，1，2）和平铺松散圈的精确最小余度条件，从而推广了Czygrinow，DeBiasio和Nagle，和Czygrinow的结果，分别。
Abstract Given two k-graphs (k-uniform hypergraphs) F and H, a perfect F-tiling (or F-factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. For all complete k-partite k-graphs K, Mycroft proved a minimum codegree condition that guarantees a K-factor in an n-vertex k-graph, which is tight up to an error term o(n). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K(k)(1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.