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

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