co.combinatorics – Number of pairs of edge-disjoint Hamilton cycles in complete graphs

Question:
how many pairs $lbrace H_i, H_jrbrace$ of edge-disjoint Hamilton cycles are in the complete graph $K_n$ with $n$ vertices?

while I could find information to the maximal number of edge-disjoint Hamilton cycles in $K_n$ I was not able to find anything about the number of combinations $lbrace H_1,,dots,,H_hrbrace$, i.e. the number of ways to select $h$ edge-disjoint Hamilton cycles from $K_n$, specifically for $h=2$

Edge-disjoint paths in trees – Computer Science Stack Exchange

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Diagrams – Maximum edge-disjoint flow

Imagine the case where you have two types of currents, say "red" current and "blue" current. You want to send $ k_r $ red river and $ k_b $ blue river through a DAG $ G $ so that no edge carries both red and blue flow. Is there an efficient way to determine if an association of $ k_r $ red and and $ k_b $ blue river exists? If so, can it be extended to several "colors"?

An extremely naive solution would be to take any subset of edges $ E & # 39; $ and check whether you can send or not $ k_r $ flow through $ G [E & # 39;] $ and $ k_b $ flow through $ G [E setminus E & # 39;] $.