Even cycle creating paths

We say that two graphs $H_1,H_2$ on the same vertex set are $G$-creating if the union of the two graphs contains $G$ as a subgraph. Let $H\left(n,k\right)$ be the maximum number of pairwise $C_k$-creating Hamiltonian paths of the complete graph $K_n$. The behavior of $H\left(n,2k+1\right)$ is much better understood than the behavior of $H\left(n,2k\right)$, the former is an exponential function of $n$ whereas the latter is larger than exponential, for every fixed $k$. We study $H\left(n,k\right)$ for fixed $k$ and $n$ tending to infinity. The only nontrivial upper bound on $H\left(n,2k\right)$ was proved by Cohen, Fachini, and Körner in the case of $k=2$: $$n^{(1/2)n-o\left(n\right)}\le H\left(n,4\right)\le n^{\left(1-1/4\right)n-o\left(n\right)}\mathrm{.}$$ In this paper, we generalize their method to prove that for every $k\ge 2$, $$n^{(1/k)n-o\left(n\right)}\le H\left(n,2k\right)\le n^{\left(1-2/(3k^2-2k)\right)n-o\left(n\right)}$$ and a similar, slightly better upper bound holds when $k$ is odd. Our proof uses constructions of bipartite, regular, $C_{2k}$-free graphs with many edges given in papers by Reiman, Benson, Lazebnik, Ustimenko, and Woldar.