Slightly later, and less formally published reference, than David E.'s citation:
Sharir, Micha, Adam Sheffer, and Emo Welzl. "Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn's technique." Proceedings 28th Symposium on Computational Geometry. ACM, 2012. (arXiv abs.)
[...] a new upper bound of $O(54.5^N)$ on the number of crossing-free straight-edge spanning cycles that can be embedded over any specific set of $N$ points in the plane, improving upon the previous best upper bound $O(68.7^N)$.
[![Fig5][1]][1]