Even circuit theorem

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(n^{1 + 1/k}) edges. For instance, 4-cycle-free graphs have O(n^3/2) edges, 6-cycle-free graphs have O(n^4/3) edges, etc.

History

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

Lower bounds

Unsolved problem in mathematics: Do there exist [math]\displaystyle{ 2k }[/math]-cycle-free graphs (for [math]\displaystyle{ k }[/math] other than [math]\displaystyle{ 2 }[/math], [math]\displaystyle{ 3 }[/math], or [math]\displaystyle{ 5 }[/math]) that have [math]\displaystyle{ \Omega(n^{1+1/k}) }[/math] edges? (more unsolved problems in mathematics)

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(n^{1 + 1/k}) edges that have no 2k-cycle.

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

Constant factors

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

[math]\displaystyle{ n^{3/2}\left(\frac{1}{2}+o(1)\right). }[/math]

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

[math]\displaystyle{ n^{1+1/k}\left(\frac{1}{2}+o(1)\right). }[/math]

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

[math]\displaystyle{ 0.5338n^{4/3} \le \operatorname{ex}(n,C_6) \le 0.6272n^{4/3}, }[/math]

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

[math]\displaystyle{ 4\left(\frac{n}{5}\right)^{6/5} \approx 0.5798 n^{6/5}. }[/math]

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

[math]\displaystyle{ n^{1+1/k}\left(k-1+o(1)\right). }[/math]