On the Turán number for the hexagon

Zoltan Füredi, Assaf Naor, Jacques Verstraëte

Research output: Contribution to journalArticle

Abstract

A long-standing conjecture of Erdo{combining double acute accent}s and Simonovits is that ex ( n, C2 k ), the maximum number of edges in an n-vertex graph without a 2 k-gon is asymptotically frac(1, 2) n1 + 1 / k as n tends to infinity. This was known almost 40 years ago in the case of quadrilaterals. In this paper, we construct a counterexample to the conjecture in the case of hexagons. For infinitely many n, we prove that{A formula is presented}We also show that ex ( n, C6 ) {less-than or slanted equal to} λ n4 / 3 + O ( n ) < 0.6272 n4 / 3 if n is sufficiently large, where λ is the real root of 16 λ3 - 4 λ2 + λ - 3 = 0. This yields the best-known upper bound for the number of edges in a hexagon-free graph. The same methods are applied to find a tight bound for the maximum size of a hexagon-free 2 n × n bipartite graph.

Original languageEnglish (US)
Pages (from-to)476-496
Number of pages21
JournalAdvances in Mathematics
Volume203
Issue number2
DOIs
StatePublished - Jul 10 2006

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Hexagon
Real Roots
Less than or equal to
Graph in graph theory
Bipartite Graph
Acute
Counterexample
Infinity
Tend
Upper bound
Vertex of a graph

Keywords

  • Excluded cycles
  • Extremal graph theory
  • Turán numbers

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Füredi, Z., Naor, A., & Verstraëte, J. (2006). On the Turán number for the hexagon. Advances in Mathematics, 203(2), 476-496. https://doi.org/10.1016/j.aim.2005.04.011

On the Turán number for the hexagon. / Füredi, Zoltan; Naor, Assaf; Verstraëte, Jacques.

In: Advances in Mathematics, Vol. 203, No. 2, 10.07.2006, p. 476-496.

Research output: Contribution to journalArticle

Füredi, Z, Naor, A & Verstraëte, J 2006, 'On the Turán number for the hexagon', Advances in Mathematics, vol. 203, no. 2, pp. 476-496. https://doi.org/10.1016/j.aim.2005.04.011
Füredi, Zoltan ; Naor, Assaf ; Verstraëte, Jacques. / On the Turán number for the hexagon. In: Advances in Mathematics. 2006 ; Vol. 203, No. 2. pp. 476-496.
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