On the maximum number of five-cycles in a triangle-free graph

  title={On the maximum number of five-cycles in a triangle-free graph},
  author={Andrzej Grzesik},
  journal={J. Comb. Theory, Ser. B},

Figures from this paper

Pentagons in triangle-free graphs

The Maximum Number of Pentagons in a Planar Graph

Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an $n$-vertex planar graph. They determined this number exactly for triangles and 4-cycles and

Subgraph densities in $K_r$-free graphs

In this paper we disprove a conjecture of Lidick´y and Murphy about the number of copies of a given graph in a K r -free graph and give an alternative general conjecture. We also prove an

A New Bound for the 2/3 Conjecture†

We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours

On the maximum number of odd cycles in graphs without smaller odd cycles

We prove that for each odd integer k ≥ 7 , every graph on n vertices without odd cycles of length less than k contains at most ( n ∕ k ) k cycles of length k . This extends the previous results on

Maximising the number of cycles in graphs with forbidden subgraphs

More about sparse halves in triangle-free graphs

One of Erdős’s conjectures states that every triangle-free graph on vertices has an induced subgraph on vertices with at most edges. We report several partial results towards this conjecture. In

Exact results for generalized extremal problems forbidding an even cycle

We determine the maximum number of copies of K s,s in a C 2 s +2 -free n -vertex graph for all integers s ≥ 2 and sufficiently large n . Moreover, for s ∈ { 2 , 3 } and any integer n we obtain the



On the number of C5's in a triangle-free graph

  • E. Györi
  • Mathematics, Computer Science
  • 1989
Etablissement d'une borne superieure du nombre de cycles de longueur 5 dans un graphe d'ordre n sans triangles, en relation avec une conjecture de P. Erdos

On 3-Hypergraphs with Forbidden 4-Vertex Configurations

The proof is a rather elaborate application of Cauchy-Schwarz-type arguments presented in the framework of flag algebras by re-proving a few known tight results about hypergraph Turan densities and significantly improving numerical bounds for several problems for which the exact value is not known yet.


1. G(n) is a graph of n vertices and G(n ; e) is a graph of n vertices and e edges. Is it true that if every induced subgraph of a G(10n) of 5n vertices has more than 2n 2 edges then our G(10n)

Flag algebras

The first application of a formal calculus that captures many standard arguments in the area of asymptotic extremal combinatorics is given by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density.

Hypergraphs Do Jump

These are the first examples of jumps for any r ≥ 3 in the interval [r!/rr, 1) and an improved upper bound for the Turán density of K4− = {123, 124, 134}: π(K4−) ≤ 0.2871.

Some problems in graph theory