Tight Hamilton cycles with high discrepancy

Source abstract

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 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.