Abstract
We develop a framework to study minimum d-degree conditions in k-uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path-cover and connecting arguments for all k and d at once, and thus sheds light on the underlying structural problems. Building on this, we show that one can study minimum d-degree conditions of k-uniform tight Hamilton cycles by focusing on the inner structure of the neighbourhoods. This reduces the matter to an Erdős–Gallai-type question for \((k-d)\)-uniform hypergraphs.
As an application, we derive two new bounds. First, we generalize a classic result of Rödl, Ruciński and Szemerédi for \(d=k-1\), and determine asymptotically best possible degree conditions for \(d = k-2\) and all \(k \geqslant 3\). This was proved independently by Polcyn, Reiher, Rödl and Schülke.
Secondly, we also provide a general upper bound of \(1-1/(2(k-d))\) for the tight Hamilton cycle d-degree threshold in k-graphs, narrowing the gap to the lower bound of \(1-1/\sqrt{k-d}\) due to Han and Zhao.
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Notes
- 1.
It is not hard to see that this is equivalent to the line graph L(G) being connected, where L(G) has vertex set E(G) and an edge ef whenever \(|e \cap f|=k-1\).
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Lang, R., Sanhueza-Matamala, N. (2021). Degree Conditions for Tight Hamilton Cycles. In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics(), vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83823-2_87
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