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Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

Part of: Graph theory

Published online by Cambridge University Press:  09 July 2019

Wei Gao ,
Jie Han  and
Yi Zhao
Wei Gao
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36830, USA. Email: [email protected]
Jie Han
Affiliation:
Department of Mathematics, University of Rhode Island, 5 Lippitt Road, Kingston, RI 02881, USA. Email: [email protected]
Yi Zhao*
Affiliation:
Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA
*
*Corresponding author. Email: [email protected]
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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.

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Secondary: 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 6 , November 2019 , pp. 840 - 870
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Supported by FAPESP (Proc. 2013/03447-6, 2014/18641-5, 2015/07869-8) and Simons Foundation Collaboration Grant # 630884.

‡

Partially supported by NSF grants DMS-1400073 and DMS 1700622.

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