Minimum degree conditions for tight Hamilton cycles
We develop a new framework to study minimum𝑑-degree conditions in𝑘-uniform hypergraphs, whichguarantee the existence of a tight Hamilton cycle. Ourmain theoreticalresult dealswith thetypical absorption,path cover and connecting arguments for all𝑘and𝑑at once, and thus sheds light on the underlying st...
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| Main Authors: | , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
11 April 2022
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| In: |
Journal of the London Mathematical Society
Year: 2022, Volume: 105, Issue: 4, Pages: 2249-2323 |
| ISSN: | 1469-7750 |
| DOI: | 10.1112/jlms.12561 |
| Online Access: | Verlag, kostenfrei, Volltext: https://doi.org/10.1112/jlms.12561 Verlag, kostenfrei, Volltext: https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12561 
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| Author Notes: | Richard Lang, Nicolás Sanhueza-Matamala |
| Summary: | We develop a new framework to study minimum𝑑-degree conditions in𝑘-uniform hypergraphs, whichguarantee the existence of a tight Hamilton cycle. Ourmain theoreticalresult dealswith thetypical absorption,path cover and connecting arguments for all𝑘and𝑑at once, and thus sheds light on the underlying struc-tural problems. Building on this, we show that one canstudy minimum𝑑-degree conditions of𝑘-uniform tightHamiltoncyclesbyfocusingontheinnerstructureoftheneighbourhoods. This reduces the matter to an Erdös–Gallai-type question for(𝑘 − 𝑑)-uniform hypergraphs,which is of independent interest. Once this frameworkis established, we can easily derive two new bounds.Firstly, we extend a classic result of Rödl, Ruciński andSzemerédi for𝑑=𝑘−1by determining asymptoticallybest possible degree conditions for𝑑=𝑘−2and all𝑘⩾3. This was proved independently by Polcyn, Reiher,Rödl and Schülke. Secondly, we provide a general upperbound of1 − 1∕(2(𝑘 − 𝑑))for the tight Hamilton cycle𝑑-degree threshold in𝑘-uniform hypergraphs, thus nar-rowing the gap to the lower bound of1−1∕√𝑘−𝑑dueto Han and Zhao |
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| Item Description: | Gesehen am 13.07.2022 |
| Physical Description: | Online Resource |
| ISSN: | 1469-7750 |
| DOI: | 10.1112/jlms.12561 |