  We say that two graphs on the same vertex set are $G$-creating if their union
(the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the
maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen,
Fachini and K\"orner proved \[n^{\frac{1}{2}n-o(n)}\leq H_n(C_4) \leq
n^{\frac{3}{4}n+o(n)}.\] In this paper we close the superexponential gap
between their lower and upper bounds by proving
\[n^{\frac{1}{2}n-\frac{1}{2}\frac{n}{\log{n}}-O(1)}\leq H_n(C_4) \leq
n^{\frac{1}{2}n+o\left(\frac{n}{\log{n}} \right)}.\] We also improve the
previously established upper bounds on $H_n(C_{2k})$ for $k>3$, and we present
a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri
on the maximum number of so-called pairwise reversing permutations. One of our
main tools is a theorem of Krivelevich, which roughly states that (certain
kinds of) good expanders contain many Hamiltonian paths.
