Editing Hypergraph (section)

===Tight cycles===
This definition is particularly used for <math>k</math>-uniform hypergraphs, where all hyperedges are of size <math>k</math>. A '''tight cycle''' of length <math>n</math> in a hypergraph <math>H</math> is a sequence of distinct vertices <math>v_1,\dots, v_n</math> such that every consecutive <math>k</math>-tuple <math>\{v_i, \dots, v_{i+k-1}\}</math> (indices modulo <math>n</math>) forms a hyperedge in <math>H</math>. 
This notion was introduced by Katona and Kierstead<ref>{{cite journal | author-link1= Gyula O. H. Katona  | first1=G. | last1=Katona |first2=H. A.|last2=Kierstead| title= Hamiltonian chains in hypergraphs |journal=Journal of Graph Theory |volume=30 | issue=3 |pages=205–212 |year=1999 |doi=10.1002/(SICI)1097-0118(199903)30:3<205::AID-JGT5>3.0.CO;2-O |url=https://doi.org/10.1002/(SICI)1097-0118(199903)30:3<205::AID-JGT5>3.0.CO;2-O}}</ref> and has since garnered considerable attention, particularly in the study of Hamiltonicity in extremal combinatorics.<ref>{{cite book | first1=Y. | last1=Zhao | chapter=Recent advances on Dirac-type problems for hypergraphs |title=Recent Trends in Combinatorics | series=The IMA Volumes in Mathematics and its Applications |pages=145–165 |year=2016 | volume=159 |doi=10.1007/978-3-319-24298-9_6 | isbn=978-3-319-24296-5 |chapter-url=https://doi.org/10.1007/978-3-319-24298-9_6}}</ref><ref>{{cite journal | author-link1=Daniela Kühn  | first1=D. | last1=Kühn |author-link2=Deryk Osthus |first2=D.|last2=Osthus| title=Hamilton cycles in graphs and hypergraphs: an extremal perspective |journal=Proceeding of the International Congress of Mathematicans, ICM  |pages=381–406 |year=2014 |isbn=9788961058070 |url=https://web.mat.bham.ac.uk/D.Osthus/icmsurvey4.pdf}}</ref> 

Rödl, Szemerédi, and Ruciński showed that every <math>n</math>-vertex <math>k</math>-uniform hypergraph <math>H</math> in which every <math>(k-1)</math>-subset of vertices is contained in at least <math>n/2 + o(n)</math> hyperedges contains a Hamilton cycle. 
This corresponds to an approximate hypergraph-extension of the celebrated [[Hamiltonian path#Bondy–Chvátal theorem| Dirac's theorem]] about Hamilton cycles in graphs.<ref>{{cite journal | author-link1= Vojtěch Rödl  | first1=V. | last1=Rödl |author-link2=Endre Szemerédi |first2=E.|last2=Szemerédi|first3=A. |last3=Ruciński |title=An approximate Dirac-type theorem for k-uniform hypergraphs |journal=Combinatorica |volume=28 |pages=229–260 |year=2008 | issue=2 |doi=10.1007/s00493-008-2295-z |url=https://doi.org/10.1007/s00493-008-2295-z}}</ref>

The maximum number of hyperedges in a (tightly) acyclic <math>k</math>-uniform hypergraph remains unknown. The best known bounds, obtained by Sudakov and Tomon<ref>{{cite journal | author-link1= Benny Sudakov  | first1=B. | last1=Sudakov |first2=I.|last2=Tomon |title=The Extremal Number of Tight Cycles |journal=International Mathematics Research Notices |volume=2022 |pages=9663–9684 |year=2022 | issue=13 |doi=10.1093/imrn/rnaa396 |url=https://doi.org/10.1093/imrn/rnaa396}}</ref>, show that every <math>n</math>-vertex <math>k</math>-uniform hypergraph with at least <math>n^{k-1+o(1)}</math> hyperedges must contain a tight cycle. This bound is optimal up to the <math>o(1)</math> error term.

An '''<math>\ell</math>-cycle''' generalizes the notion of a tight cycle.
It consists in a sequence of vertices <math>v_1,\dots, v_n </math> and hyperedges <math>e_1,\dots, e_t</math> where each <math>e_i</math> consists of <math>k</math> consecutive vertices in the sequence and <math>|e_i\cap e_{i+1}|=\ell</math> for every <math>1\leq i\leq t</math>. Since every edge of the <math>\ell</math>-cycle contains exactly <math>k-\ell</math> vertices which are not contained in the previous edge, <math>n</math> must be divisible by <math>k-\ell</math>. Note that <math>\ell=k-1</math> recovers the definition of a tight cycle.