Pairs of Pairs

Consider any way to pair the integers $1$ through $2N$ into $N$ pairs $(A_1,B_1), (A_2,B_2), \ldots, (A_N,B_N)$ in such a way that each of these integers is present in exactly one pair, $A_i \lt B_i$ for each valid $i$ and $A_1 \lt A_2 \lt \ldots \lt A_N$. Let's call such a sequence of pairs a *pairing*. For a pairing $(A_1,B_1), (A_2,B_2), \ldots, (A_N,B_N)$, we can create an undirected graph