Tight Hamilton cycles with high discrepancy
arxiv(2023)
摘要
In this paper, we initiate the study of discrepancy questions for spanning
subgraphs of $k$-uniform hypergraphs. Our main result is that any $2$-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 $n/2+\Omega(n)$ 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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