\section{Missing details from Section~\ref{sec:hard}}
\label{app:hard}

\subsection{Proofs of~\cref{eq:1e}-\cref{eq:2pd-3d}}
\label{app:easy-counts}

\cref{eq:1e},~\cref{eq:2p} and~\cref{eq:3s} are immediate.

A quick argument to why \cref{eq:2m} is true.  Note that for edge $(i, j)$ connecting arbitrary vertices $i$ and $j$, finding all other edges in $G$ disjoint to $(i, j)$ is equivalent to finding all edges that are not connected to either vertex $i$ or $j$.  The number of such edges is $m - d_i - d_j + 1$, where we add $1$ since edge $(i, j)$ is removed twice when subtracting both $d_i$ and $d_j$.  Since the summation is iterating over all edges such that a pair $\left((i, j), (k, \ell)\right)$ will also be counted as $\left((k, \ell), (i, j)\right)$, division by $2$ then eliminates this double counting.

\cref{eq:2pd-3d} is true for similar reasons.  For edge $(i, j)$, it is necessary to find two additional edges, disjoint or connected.  As in ~\cref{eq:2m}, once the number of edges disjoint to $(i, j)$ have been computed, then we only need to consider all possible combinations of two edges from the set of disjoint edges, since it doesn't matter if the two edges are connected or not.    Note, the factor $3$ of $\threedis$ is necessary to account for the triple counting of $3$-matchings.  It is also the case that, since the two path in $\twopathdis$ is connected, that there will be no double counting by the fact that the summation automatically 'disconnects' the current edge, meaning that a two matching at the current vertex will not be counted.  The sum over all such edge combinations is precisely then $\numocc{G}{\twopathdis} + 3\numocc{G}{\threedis}$.
