Minimum degree conditions for tight Hamilton cycles - Lang, Richard; Sanhueza-Matamala, Nicolás (Journal of the London Mathematical Society, 2022)
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