The maximum number of copies of an even cycle in a planar graph

We resolve a conjecture of Cox and Martin by determining asymptotically for every k ≥ 2 the maximum number of copies of C 2k in an n-vertex planar graph. IntroductionA fundamental problem in extremal combinatorics is maximizing the number of occurrences of subgraphs of a certain type among all graphs from a given class.In the case of n-vertex planar graphs, Hakimi and Schmeichel 8 determined the maximum possible number of cycles length 3 and 4 exactly and showed that for any k ≥ 3, the maximum number of k-cycles is Θ(n ⌊k/2⌋ ).Moreover, they proposed a conjecture for the maximum number of 5-cycles in an n-vertex planar graph which was verified much later by Győri et al. in 6.The maximum number of 6-cycles and 8-cycles was settled asymptotically by Cox and Martin in 3, and later the same authors 4 also determined the maximum number of 10-cycles and 12-cycles asymptotically.Following the work of Hakimi and Schmeichel 8, Alon and Caro 1 considered the general problem of maximizing copies of a given graph H among n-vertex planar graphs.Wormald 11 and later independently Eppstein 5 showed that for 3-connected H, the maximum number of copies of H is Θ(n).The order of magnitude in the case when H is a tree was determined in 7, and the order of magnitude for an arbitrary graph was settled by Huynh, Joret and Wood 9.Note that by Kuratowski's theorem 10 such problems can be thought of as generalized Turán problems where we maximize the number of copies of the graph H while forbidding all subdivisions of K 5 and K 3,3 .Given that the order of magnitude of the maximum number of copies of any graph H in an n-vertex planar graph is determined, it is natural to look for sharp asymptotic results.While in recent times a number of results have been obtained about the asymptotic number of H-copies in several specific cases, less is known for general classes of graphs.Cox and Martin 3 introduced some general tools for studying such problems and conjectured that in the case of an even cycle C 2k with k ≥ 3, the maximum number of copies is asymptotically n k /k k .We confirm their conjecture.