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arXiv · 2002.12594

A discrepancy version of the Hajnal-Szemer\'edi theorem

József Balogh·Béla Csaba·András Pluhár·Andrew Treglown
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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--Szemer\'edi 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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József Balogh, Béla Csaba, András Pluhár, Andrew Treglown. 2020-02-28. A discrepancy version of the Hajnal-Szemer\'edi theorem. https://arxiv.org/abs/2002.12594

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