A Tight Bound for Minimal Connectivity

TL;DR

This paper improves the bounds on the number of degree-k vertices in minimally k-connected graphs by proposing a new, tight bound that extends Mader's and surpasses Oxley's in certain cases.

Contribution

It introduces a new tight bound for minimally k-connected graphs that generalizes Mader's bound and improves upon Oxley's previous results.

Findings

The new bound is proven to be the best possible.

It coincides with Mader's bound for certain parameters.

The bound is tighter than Oxley's for small edge counts.

Abstract

For minimally $k$-connected graphs on $n$ vertices, Mader proved a tight lower bound for the number $|V_k|$ of vertices of degree $k$ in dependence on $n$ and $k$. Oxley observed 1981 that in many cases a considerably better bound can be given if $m := |E|$ is used as additional parameter, i.e. in dependence on $m$, $n$ and $k$. It was left open to determine whether Oxley's bound is best possible. We show that this is not the case, but propose a closely related bound that deviates from Oxley's long-standing one only for small values of $m$. We prove that this new bound is best possible. The bound contains Mader's bound as special case.