TY - JOUR

T1 - Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

AU - Gao, Wei

AU - Han, Jie

AU - Zhao, Yi

N1 - Publisher Copyright:
© Cambridge University Press 2019.

PY - 2019/11/1

Y1 - 2019/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85068840571&partnerID=8YFLogxK

U2 - 10.1017/S096354831900021X

DO - 10.1017/S096354831900021X

M3 - Article

AN - SCOPUS:85068840571

SN - 0963-5483

VL - 28

SP - 840

EP - 870

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 6

ER -