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A discrepancy version of the Hajnal–Szemerédi theorem

Part of: Graph theory

Published online by Cambridge University Press:  30 October 2020

József Balogh ,
Béla Csaba ,
András Pluhár  and
Andrew Treglown
József Balogh
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, IL61801, USA, and Moscow Institute of Physics and Technology, Dolgoprudny, Russian Federation
Béla Csaba
Affiliation:
Bolyai Institute, University of Szeged, Hungary
András Pluhár
Affiliation:
Department of Computer Science, University of Szeged, Hungary
Andrew Treglown*
Affiliation:
University of Birmingham, Edgbaston, B15 2TT, UK
*
*Corresponding author. Email: a.c.treglown@bham.ac.uk
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Abstract

A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-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 Kr-tiling of high discrepancy.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C70: Factorization, matching, partitioning, covering and packing
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 3 , May 2021 , pp. 444 - 459
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Research of this author is partially supported by NSF grants DMS-1500121, DMS-1764123, Arnold O. Beckman Research Award (UIUC) Campus Research Board 18132 and the Langan Scholar Fund (UIUC).

‡

Research of this author was supported in part by the grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary, and by NKFIH grant KH_18 129597.

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