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arXiv · 2312.09976

Tight Hamilton cycles with high discrepancy

Lior Gishboliner·Stefan Glock·Amedeo Sgueglia
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Abstract

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$ with minimum $(k-1)$-degree $\delta(G) \ge (1/2+o(1))n$ contains a tight Hamilton cycle with high discrepancy, that is, with at least $n/r+\Omega(n)$ edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Tur\'an-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.

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BibTeXRIS

Lior Gishboliner, Stefan Glock, Amedeo Sgueglia. 2023-12-15. Tight Hamilton cycles with high discrepancy. https://doi.org/10.1017/s0963548325000057

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