Let $G_{n,m} = (V,E)$ be a complete graph on $n$ vertices, $n \in \mathbb{Z}^+$, with one additional feature: there are $m$ loops at every vertex, $m \in \mathbb{N}$. In other words, for every vertex $u$ and for every vertex $v$ in $V$, if $u \neq v$, then there is exactly one $\{u,v\}\in E$, and we have $\{v,v\}_i \in E$ for $i=1,2,...,m$.
I am trying to determine the number of edge coverings on such a graph, which I denote as $C(G_{n,m})$. In other words, how many subsets of $E$ exist in which every vertex in $V$ is incident with at least one edge?
I stumbled upon a paper conveying the edge coverings of a complete graph on $n$ vertices, denoted as $K_n$, as
$$C(K_n) = \sum_{j=0}^{n} (-1)^j \binom{n}{j} 2^{\binom{n-j}{2}}$$
It is also given for $k$ edges, $0 \le k \le \binom{n}{2}$,
$$C(K_n,k) = \sum_{j=0}^{n} (-1)^j \binom{n}{j} {\binom{\binom{n-j}{2}}{k}}$$
but I am still having difficulty given the graph I'm examining contains loops as well. My best attempt has consisted of subtracting from the powerset of $E$ all those subsets in which there is at least $1$ isolated vertex. I ended up with a recursive equation as follows:
$$ C(G_{n,m}) = 2^{\binom{n}{2}+nm} - 1 - \sum_{t=1}^{n-1} \binom{n}{t} C(G_{n-t,m})$$
Any help is appreciated. Thanks!
Notice, first, that the first formula is an inclusion-exclusion one, so you can not just take out the elements without including the repetition back. In that way, call $$A_i = \{\text{collection of edges not covering vertex }i\},$$ so $$|A_i|=2^{\binom{n-1}{2}+n\cdot m-m},$$ and so, if you want to get the edges that do not cover any vertex in a set $X,$ you obtain $$\left | \bigcap _{x\in X} A_x\right |=2^{\binom{n-|X|}{2}+n\cdot m-|X|\cdot m},$$ because you are taking away the edges connected to $x\in X$ and each one has $m$ loops that you have now to take away. Then you are interested in $$\left |\mathcal{P}(E)\setminus \bigcup _{v\in V}A_v\right |=2^{\binom{n}{2}+n\cdot m}-\left | \bigcup _{v\in V}A_v\right |$$ using PIE and the formula we found for intersection, we get that $$C(G_{n,m})=\sum_{j=0}^{n} (-1)^j \binom{n}{j} 2^{\binom{n-j}{2}+(n-j)\cdot m}.$$