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Tight Hamilton cycles with high discrepancy
- Part of
-
- Journal:
- Combinatorics, Probability and Computing / Volume 34 / Issue 4 / July 2025
- Published online by Cambridge University Press:
- 30 May 2025, pp. 565-584
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- Article
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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 Turán-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.
