Minimum degree conditions for tight Hamilton cycles

• Verfasst von: Lang, Richard [VerfasserIn]
• Verfasst von: Sanhueza-Matamala, Nicolás [VerfasserIn]
• Titel: Minimum degree conditions for tight Hamilton cycles
• Verf.angabe: Richard Lang, Nicolás Sanhueza-Matamala
• E-Jahr: 2022
• Jahr: 11 April 2022
• Umfang: 75 S.
• Fussnoten: Gesehen am 13.07.2022
• Titel Quelle: Enthalten in: London Mathematical SocietyJournal of the London Mathematical Society
• Ort Quelle: Oxford : Wiley, 1926
• Jahr Quelle: 2022
• Band/Heft Quelle: 105(2022), 4, Seite 2249-2323
• ISSN Quelle: 1469-7750
• DOI: doi:10.1112/jlms.12561
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1810103193
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

We develop a new framework to study minimum????-degree conditions in????-uniform hypergraphs, whichguarantee the existence of a tight Hamilton cycle. Ourmain theoreticalresult dealswith thetypical absorption,path cover and connecting arguments for all????and????at once, and thus sheds light on the underlying struc-tural problems. Building on this, we show that one canstudy minimum????-degree conditions of????-uniform tightHamiltoncyclesbyfocusingontheinnerstructureoftheneighbourhoods. This reduces the matter to an Erdös–Gallai-type question for(???? − ????)-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 for????=????−1by determining asymptoticallybest possible degree conditions for????=????−2and all????⩾3. This was proved independently by Polcyn, Reiher,Rödl and Schülke. Secondly, we provide a general upperbound of1 − 1∕(2(???? − ????))for the tight Hamilton cycle????-degree threshold in????-uniform hypergraphs, thus nar-rowing the gap to the lower bound of1−1∕√????−????dueto Han and Zhao