Even circuit theorem
# 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 (1974) 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

Do there exist $2k$-cycle-free graphs (for $k$ other than $2$, $3$, or $5$) that have $\Omega (n^{1+1/k})$ 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 $Ω(n^{1+1/k})$ 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

    $n^{3/2}\left({\frac {1}{2}}+o(1)\right).$

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

    $n^{1+1/k}\left({\frac {1}{2}}+o(1)\right).$

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

    $0.5338n^{4/3}\leq \operatorname {ex} (n,C_{6})\leq 0.6272n^{4/3},$

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

    $4\left({\frac {n}{5}}\right)^{6/5}\approx 0.5798n^{6/5}.$

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

    $n^{1+1/k}\left(k-1+o(1)\right).$
