  A simple graph $G$ with edge-connectivity $\lambda(G)$ and minimum degree
$\delta(G)$ is maximally edge connected if $\lambda(G)=\delta(G)$. In 1964,
given a non-increasing degree sequence $\pi=(d_{1},\ldots,d_{n})$, Jack Edmonds
showed that there is a realization $G$ of $\pi$ that is $k$-edge-connected if
and only if $d_{n}\geq k$ with $\sum_{i=1}^{n}d_{i}\geq 2(n-1)$ when $d_{n}=1$.
We strengthen Edmonds's result by showing that given a realization $G_{0}$ of
$\pi$ if $Z_{0}$ is a spanning subgraph of $G_{0}$ with $\delta(Z_{0})\geq 1$
such that $|E(Z_{0})|\geq n-1$ when $\delta(G_{0})=1$, then there is a
maximally edge-connected realization of $\pi$ with $G_{0}-E(Z_{0})$ as a
subgraph. Our theorem tells us that there is a maximally edge-connected
realization of $\pi$ that differs from $G_{0}$ by at most $n-1$ edges. For
$\delta(G_{0})\geq 2$, if $G_{0}$ has a spanning forest with $c$ components,
then our theorem says there is a maximally edge-connected realization that
differs from $G_{0}$ by at most $n-c$ edges. As an application we combine our
work with Kundu's $k$-factor Theorem to show there is a maximally
edge-connected realization with a $(k_{1},\dots,k_{n})$-factor for $k\leq
k_{i}\leq k+1$ and present a partial result to a conjecture that strengthens
the regular case of Kundu's $k$-factor theorem.
