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

Abstract

A simple graph $G$ with edge-connectivity $\lambda \left(G\right)$ and minimum degree $\delta \left(G\right)$ is maximally edge-connected if $\lambda \left(G\right)=\delta \left(G\right)$. In 1964, given a nonincreasing degree sequence $\pi =\left(d_1,\dots ,d_n\right)$, Jack Edmonds showed that there is a realization $G$ of $\pi$ that is $k$-edge-connected if and only if $d_n\ge k$ with ${\sum }_{i=1}^n{d}_i\ge 2\left(n-1\right)$ 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 $\delta \left(Z_0\right)\ge 1$ such that $\mid E\left(Z_0\right)\mid \ge n-1$ when $\delta \left(G_0\right)=1$, then there is a maximally edge-connected realization of $\pi$ with $G_0-E\left(Z_0\right)$ 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 $\delta \left(G_0\right)\ge 2$, 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 $\left(k_1,\dots ,k_n\right)$-factor for $k\le k_i\le k+1$ and present a partial result to a conjecture that strengthens the regular case of Kundu's $k$-factor theorem.