Hamilton l-cycles in uniform hypergraphs

Abstract

We say that a k-uniform hypergraph C is an l-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely l vertices. We prove that if 1 \leq l \leq k-1 and k-l does not divide k then any k-uniform hypergraph on n vertices with minimum degree at least n/((\lceil (k/(k-l)) \rceil)(k-l))+o(n) contains a Hamilton l-cycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an l-cycle for any l with 1 \leq l \leq k-1.

v3: corrected very minor error in Lemma 4.6 and the proof of Lemma 6.2

References in corpus (3)

• Loose Hamilton cycles in hypergraphs
• Hypergraph regularity and the multidimensional Szemerédi theorem
• A hypergraph blow-up lemma

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