The maximum number of 10-and 12-cycles in a planar graph

For a fixed planar graph H, let NP(n, H) denote the maximum number of copies of H in an n-vertex planar graph. In the case when H is a cycle, the asymptotic value of NP(n, Cm) is currently known for m is an element of {3, 4, 5, 6, 81. In this note, we extend this list by establishing NP(n, C10) -(n/5)5 and NP(n, C12) -(n/6)6. We prove this by answering the following question for m is an element of {5, 61, which is interesting in its own right: which probability mass mu on the edges of some clique maximizes the probability that m independent samples from mu form an m-cycle?(c) 2022 Elsevier B.V. All rights reserved.