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Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC2k, the cycle of length 2k, isO(n1+1/k). Simonovits established a corresponding supersaturation result forC2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn1+1/kcontains at leastc(e(G)/v(G))2kcopies ofC2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.
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This content will become publicly available on January 17, 2026
A Hypergraph Analog of Dirac's Theorem for Long Cycles in 2-Connected Graphs, II: Large Uniformities
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 $$k \geq r+2$$. In this paper we address the case $$k \leq r+1$$ 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 $$|E(H)|\geq n$$, this bound coincides with the bound of the Dirac's Theorem for 2-connected graphs.
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- PAR ID:
- 10592107
- Publisher / Repository:
- Electronic Journal of Combinatorics
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 32
- Issue:
- 1
- ISSN:
- 1077-8926
- Subject(s) / Keyword(s):
- Berge cycles hypergraphs
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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