2023/10/19 by Alexandr Kostochka, Kostochka, Alexandr, Ruth Luo +3
Engineering · Mathematics · #05C35 #05C38 #05C65 #05D05 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2310.13190
openalex publication_date 2023/10/19 · openalex created_date 2023/10/24 · openalex updated_date 2026/07/28
Dirac proved that each n-vertex 2-connected graph with minimum degree k contains a cycle of length at least min\2k, n\. We obtain analogous results for Berge cycles in hypergraphs. Recently, the authors proved an exact lower bound on the minimum degree ensuring a Berge cycle of length at least min\2k, n\ in n-vertex r-uniform 2-connected hypergraphs when k ≥ r+2. In this paper we address the case k ≤ r+1 in which the bounds have a different behavior. We prove that each n-vertex r-uniform 2-connected hypergraph H with minimum degree k contains a Berge cycle of length at least min\2k,n,|E(H)|\. If |E(H)|≥ n, this bound coincides with the bound of the Dirac's Theorem for 2-connected graphs.