Extremal problems on saturation for the family of $k$-edge-connected graphs

Abstract

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E\left(\overline{G}\right)$. The saturation number and extremal number of $\mathcal{F}$, denoted $\mathrm{sat}(n,\mathcal{F})$ and $\mathrm{ex}(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in \mathbb{N}$, let ${\mathcal{F}}_k$ and ${\mathcal{F}}_k^{\prime }$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $\mathrm{sat}(n,{\mathcal{F}}_k)=(k-1)n-\left(\genfrac{}{}{0ex}{}{k}{2}\right)$; we prove $\mathrm{sat}(n,{\mathcal{F}}_k^{\prime })=(k-1)(n-1)-⌊\frac{n}{k+1}⌋\left(\genfrac{}{}{0ex}{}{k-1}{2}\right)$. We also prove $\mathrm{ex}(n,{\mathcal{F}}_k^{\prime })=(k-1)n-\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ and characterize when equality holds. Finally, we give a lower bound on the spectral radius for ${\mathcal{F}}_k$-saturated and ${\mathcal{F}}_k^{\prime }$-saturated graphs.