Bounding \(\ell\)-edge-connectivity in edge-connectivity (Q2172417)

This paper studies bounding \(\ell\)-edge-connectivity in edge-connectivity. For a connected graph \(G\) and \(\ell\ge2\), \(\kappa^\prime(G)\) is the edge-connectivity and \(\kappa^\prime_{\ell}(G)\) is the \(\ell\)-edge-connectivity. For any integer \(k\ge1\) and a family \(\mathcal{F}\) of connected graphs with order at least \(\ell\), define \(\Phi(\ell,k,\mathcal{F})=\sup\{\kappa^\prime_{\ell}(G):G\in\mathcal{F},\kappa^\prime(G)=k\}\) and \(\phi(\ell,k,\mathcal{F})=\inf\{\kappa^\prime_{\ell}(G):G\in\mathcal{F},\kappa^\prime (G)=k\}\). Let \(\mathcal{G}^u\) be the family of all edge-uniformly dense graphs of order at least \(\ell\). It is shown that \(\Phi(\ell,k,\mathcal{G}^u)=(\ell-1)k\) and \(\phi(\ell,k,\mathcal{G}^u)=k\ell/2\).