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Even circuit theorem

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Even circuit theorem
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In extremal graph theory, the even circuit theorem is a result of Paul Erdős according to which an n-vertex graph that does not have a simple cycle of length 2k can only have O(n1 + 1/k) edges. For instance, 4-cycle-free graphs have O(n3/2) edges, 6-cycle-free graphs have O(n4/3) edges, etc.

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The highest amount of edges for 7 vertices banning 4 and 6 cycles respectively

History

The result was stated without proof by Erdős in 1964.[1] Bondy & Simonovits (1974) published the first proof, and strengthened the theorem to show that, for n-vertex graphs with Ω(n1 + 1/k) edges, all even cycle lengths between 2k and 2kn1/k occur.[2]

Lower bounds

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Perspective
Unsolved problem in mathematics
Do there exist -cycle-free graphs (for other than , , or ) that have edges?

The bound of Erdős's theorem is tight up to constant factors for some small values of k: for k = 2, 3, or 5, there exist graphs with Ω(n1 + 1/k) edges that have no 2k-cycle.[2][3][4]

It is unknown for k other than 2, 3, or 5 whether there exist graphs that have no 2k-cycle but have Ω(n1 + 1/k) edges, matching Erdős's upper bound.[5] Only a weaker bound is known, according to which the number of edges can be Ω(n1 + 2/(3k 3)) for odd values of k, or Ω(n1 + 2/(3k 4)) for even values of k.[4]

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Constant factors

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Perspective

Because a 4-cycle is a complete bipartite graph, the maximum number of edges in a 4-cycle-free graph can be seen as a special case of the Zarankiewicz problem on forbidden complete bipartite graphs, and the even circuit theorem for this case can be seen as a special case of the Kővári–Sós–Turán theorem. More precisely, in this case it is known that the maximum number of edges in a 4-cycle-free graph is

Erdős & Simonovits (1982) conjectured that, more generally, the maximum number of edges in a 2k-cycle-free graph is

[6]

However, later researchers found that there exist 6-cycle-free graphs and 10-cycle-free graphs with a number of edges that is larger by a constant factor than this conjectured bound, disproving the conjecture. More precisely, the maximum number of edges in a 6-cycle-free graph lies between the bounds

where ex(n,G) denotes the maximum number of edges in an n-vertex graph that has no subgraph isomorphic to G.[3] The maximum number of edges in a 10-cycle-free graph can be at least[4]

The best proven upper bound on the number of edges, for 2k-cycle-free graphs for arbitrary values of k, is

[5]
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References

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