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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.
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A discrepancy version of the Hajnal–Szemerédi theorem
Abstract A perfect K r -tiling in a graph G is a collection of vertex-disjoint copies of the clique K r in G covering every vertex of G . The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect K r -tiling in a graph G . The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks substructures F of G that have ‘high’ discrepancy ( i.e. the sum of the labels of the edges in F is far from 0). In this paper we determine the minimum degree threshold for a graph to contain a perfect K r -tiling of high discrepancy.
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- Award ID(s):
- 1764123
- PAR ID:
- 10225051
- Date Published:
- Journal Name:
- Combinatorics, Probability and Computing
- Volume:
- 30
- Issue:
- 3
- ISSN:
- 0963-5483
- Page Range / eLocation ID:
- 444 to 459
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0963548320000516
