
arXiv: 2410.21733
An n n -vertex graph G G is pancyclic if for every ℓ \ell , 3 ≤ ℓ ≤ n 3\leq \ell \leq n , G G contains a cycle of length ℓ \ell . We consider an analog for hypergraphs. An r r -uniform hypergraph is a set of vertices and a set of edges in which every edge contains r r vertices. A Berge cycle of length ℓ \ell in a hypergraph is an alternating sequence of ℓ \ell distinct vertices and ℓ \ell distinct edges v 1 , e 1 , v 2 , … , v ℓ , e ℓ , v 1 v_1,e_1,v_2, \ldots , v_\ell , e_{\ell },v_1 such that { v i , v i + 1 } ⊆ e i \{v_i, v_{i+1}\} \subseteq e_i for all i i , with indices taken modulo ℓ \ell . We similarly call a hypergraph Berge pancyclic if it contains Berge cycles of every possible length. For sufficiently large n n we prove a sharp minimum degree condition guaranteeing Berge pancyclicity in r r -uniform hypergraphs where 3 ≤ r ≤ 3 \leq r \leq ⌊ ( n − 1 ) / 2 ⌋ − 2 \lfloor (n-1)/2\rfloor - 2 . This bound is optimal for all such combinations of n n and r r .
FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
| selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | 0 | |
| popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network. | Average | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Average | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Average |
