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A simple graphG    $G$    with edge‐connectivityλ   (  G  )$\lambda (G)$    and minimum degreeδ   (  G  )$\delta (G)$    is maximally edge‐connected ifλ   (  G  )   =  δ   (  G  )$\lambda (G)=\delta (G)$   . In 1964, given a nonincreasing degree sequenceπ  =   (    d  1   ,  …  ,   d  n    )$\pi =({d}_{1},{\rm{\ldots }},{d}_{n})$   , Jack Edmonds showed that there is a realizationG    $G$    ofπ    $\pi $    that isk    $k$   ‐edge‐connected if and only ifd  n   ≥  k    ${d}_{n}\ge k$    with∑   i  =  1   n    d  i   ≥  2   (   n  −  1   )${\sum }_{i=1}^{n}{d}_{i}\ge 2(n-1)$    whend  n   =  1    ${d}_{n}=1$   . We strengthen Edmonds's result by showing that given a realizationG  0${G}_{0}$    ofπ    $\pi $    ifZ  0${Z}_{0}$    is a spanning subgraph ofG  0${G}_{0}$    withδ   (   Z  0   )   ≥  1    $\delta ({Z}_{0})\ge 1$    such that∣  E   (   Z  0   )   ∣  ≥  n  −  1    $| E({Z}_{0})| \ge n-1$    whenδ   (   G  0   )   =  1    $\delta ({G}_{0})=1$   , then there is a maximally edge‐connected realization ofπ    $\pi $    withG  0   −  E   (   Z  0   )${G}_{0}-E({Z}_{0})$    as a subgraph. Our theorem tells us that there is a maximally edge‐connected realization ofπ    $\pi $    that differs fromG  0${G}_{0}$    by at mostn  −  1    $n-1$    edges. Forδ   (   G  0   )   ≥  2    $\delta ({G}_{0})\ge 2$   , ifG  0${G}_{0}$    has a spanning forest withc    $c$    components, then our theorem says there is a maximally edge‐connected realization that differs fromG  0${G}_{0}$    by at mostn  −  c    $n-c$    edges. As an application we combine our work with Kundu'sk    $k$   ‐factor theorem to show there is a maximally edge‐connected realization with a(    k  1   ,  …  ,   k  n    )$({k}_{1},{\rm{\ldots }},{k}_{n})$   ‐factor fork  ≤   k  i   ≤  k  +  1    $k\le {k}_{i}\le k+1$    and present a partial result to a conjecture that strengthens the regular case of Kundu'sk    $k$   ‐factor theorem.

Maximally edge‐connected realizations and Kundu's k $k$ ‐factor theorem