Maximally Edge-Connected Realizations and Kundu's $k$-factor Theorem

Abstract

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.