We study the function $H_n(C_{2k})$, the maximum number of Hamilton paths such that the union of any pair of them contains $C_{2k}$ as a subgraph. We give upper bounds on this quantity for $k\ge 3$, improving results of Harcos and Soltész, and we show that if a conjecture of Ustimenko is true then one additionally obtains improved upper bounds for all $k\geq 6$. {We also give bounds on $H_n(K_{2,3})$ and $H_n(K_{2,4})$. In order to prove our results, we extend a theorem of Krivelevich which counts Hamilton cycles in $(n, d, λ)$-graphs to bipartite or irregular graphs, and then apply these results to generalized polygons and the constructions of Lubotzky-Phillips-Sarnak and Füredi.