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.