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.
Contents
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(n1 + 1/k) edges, all even cycle lengths between 2k and 2kn1/k occur.
Lower bounds
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.
It is an 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. 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.
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
Erdős & Simonovits (1982) conjectured that, more generally, the maximum number of edges in a 2k-cycle-free graph is
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. The maximum number of edges in a 10-cycle-free graph can be at least
The best proven upper bound on the number of edges, for 2k-cycle-free graphs for arbitrary values of k, is