Odd cycles and Θ-cycles in hypergraphs

A. Gyárfás, Michael S. Jacobson, André E. Kézdy, Jeno Lehel

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 - 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.

Original languageEnglish
Pages (from-to)2481-2491
Number of pages11
JournalDiscrete Mathematics
Volume306
Issue number19-20
DOIs
Publication statusPublished - Oct 6 2006

Fingerprint

Odd Cycle
Hypergraph
Cycle
Uniform Hypergraph
Hypertree
Imply
Minimum Degree
Doubling
Chord or secant line
Bipartite Graph
Paul Adrien Maurice Dirac
Connected graph
Equality
If and only if

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Gyárfás, A., Jacobson, M. S., Kézdy, A. E., & Lehel, J. (2006). Odd cycles and Θ-cycles in hypergraphs. 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.

In: Discrete Mathematics, Vol. 306, No. 19-20, 06.10.2006, p. 2481-2491.

Research output: Contribution to journalArticle

Gyárfás, A, Jacobson, MS, Kézdy, AE & Lehel, J 2006, 'Odd cycles and Θ-cycles in hypergraphs', Discrete Mathematics, vol. 306, no. 19-20, pp. 2481-2491. https://doi.org/10.1016/j.disc.2005.12.037
Gyárfás, A. ; Jacobson, Michael S. ; Kézdy, André E. ; Lehel, Jeno. / Odd cycles and Θ-cycles in hypergraphs. In: Discrete Mathematics. 2006 ; Vol. 306, No. 19-20. pp. 2481-2491.
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