### Abstract

A Θ-cycle of a hypergraph is a cycle including an edge that contains at least three base points of the cycle. We show that if a hypergraph H = (V, E) has no Θ-cycle, and | e | ≥ 3, for every edge e ∈ E, then ∑_{e ∈ E} (| e | - 1) ≤ 2 | V | - 2 with equality if and only if H is obtained from a hypertree by doubling its edges. This result reminiscent of Berge's and Lovász's similar inequalities implies that 3-uniform hypergraphs with n vertices and n edges have Θ-cycles, and 3-uniform simple hypergraphs with n vertices and n - 1 edges have Θ-cycles. Both results are sharp. Since the presence of a Θ-cycle implies the presence of an odd cycle, both results are sharp for odd cycles as well. However, for linear 3-uniform hypergraphs the thresholds are different for Θ-cycles and for odd cycles. Linear 3-uniform hypergraphs with n vertices and with minimum degree two have Θ-cycles when | E | ≥ 5 n / 6 - c_{1} sqrt(n) and have odd cycles when | E | ≥ 7 n / 9 - c_{2} sqrt(n) and these are sharp results apart from the values of the constants. Most of our proofs use the concept of edge-critical (minimally 2-connected) graphs introduced by Dirac and by Plummer. In fact, the hypergraph results-in disguise-are extremal results for bipartite graphs that have no cycles with chords.

Original language | English |
---|---|

Pages (from-to) | 2481-2491 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 306 |

Issue number | 19-20 |

DOIs | |

Publication status | Published - Oct 6 2006 |

### Fingerprint

### Keywords

- Bipartite graph
- Extremal problem
- Hypergraph
- Minimal blocks
- Odd cycles

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*306*(19-20), 2481-2491. https://doi.org/10.1016/j.disc.2005.12.037

**Odd cycles and Θ-cycles in hypergraphs.** / Gyárfás, A.; Jacobson, Michael S.; Kézdy, André E.; Lehel, Jeno.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 306, no. 19-20, pp. 2481-2491. https://doi.org/10.1016/j.disc.2005.12.037

}

TY - JOUR

T1 - Odd cycles and Θ-cycles in hypergraphs

AU - Gyárfás, A.

AU - Jacobson, Michael S.

AU - Kézdy, André E.

AU - Lehel, Jeno

PY - 2006/10/6

Y1 - 2006/10/6

N2 - A Θ-cycle of a hypergraph is a cycle including an edge that contains at least three base points of the cycle. We show that if a hypergraph H = (V, E) has no Θ-cycle, and | e | ≥ 3, for every edge e ∈ E, then ∑e ∈ E (| e | - 1) ≤ 2 | V | - 2 with equality if and only if H is obtained from a hypertree by doubling its edges. This result reminiscent of Berge's and Lovász's similar inequalities implies that 3-uniform hypergraphs with n vertices and n edges have Θ-cycles, and 3-uniform simple hypergraphs with n vertices and n - 1 edges have Θ-cycles. Both results are sharp. Since the presence of a Θ-cycle implies the presence of an odd cycle, both results are sharp for odd cycles as well. However, for linear 3-uniform hypergraphs the thresholds are different for Θ-cycles and for odd cycles. Linear 3-uniform hypergraphs with n vertices and with minimum degree two have Θ-cycles when | E | ≥ 5 n / 6 - c1 sqrt(n) and have odd cycles when | E | ≥ 7 n / 9 - c2 sqrt(n) and these are sharp results apart from the values of the constants. Most of our proofs use the concept of edge-critical (minimally 2-connected) graphs introduced by Dirac and by Plummer. In fact, the hypergraph results-in disguise-are extremal results for bipartite graphs that have no cycles with chords.

AB - A Θ-cycle of a hypergraph is a cycle including an edge that contains at least three base points of the cycle. We show that if a hypergraph H = (V, E) has no Θ-cycle, and | e | ≥ 3, for every edge e ∈ E, then ∑e ∈ E (| e | - 1) ≤ 2 | V | - 2 with equality if and only if H is obtained from a hypertree by doubling its edges. This result reminiscent of Berge's and Lovász's similar inequalities implies that 3-uniform hypergraphs with n vertices and n edges have Θ-cycles, and 3-uniform simple hypergraphs with n vertices and n - 1 edges have Θ-cycles. Both results are sharp. Since the presence of a Θ-cycle implies the presence of an odd cycle, both results are sharp for odd cycles as well. However, for linear 3-uniform hypergraphs the thresholds are different for Θ-cycles and for odd cycles. Linear 3-uniform hypergraphs with n vertices and with minimum degree two have Θ-cycles when | E | ≥ 5 n / 6 - c1 sqrt(n) and have odd cycles when | E | ≥ 7 n / 9 - c2 sqrt(n) and these are sharp results apart from the values of the constants. Most of our proofs use the concept of edge-critical (minimally 2-connected) graphs introduced by Dirac and by Plummer. In fact, the hypergraph results-in disguise-are extremal results for bipartite graphs that have no cycles with chords.

KW - Bipartite graph

KW - Extremal problem

KW - Hypergraph

KW - Minimal blocks

KW - Odd cycles

UR - http://www.scopus.com/inward/record.url?scp=33748748803&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748748803&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2005.12.037

DO - 10.1016/j.disc.2005.12.037

M3 - Article

AN - SCOPUS:33748748803

VL - 306

SP - 2481

EP - 2491

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 19-20

ER -