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

TL;DR

This paper investigates extremal and saturation properties of $k$-edge-connected graphs, providing exact formulas for saturation and extremal numbers, characterizations of equality cases, and bounds on spectral radius.

Contribution

It derives new formulas for the saturation number of $k$-edge-connected graphs and characterizes extremal graphs, extending previous results on $k$-connected graphs.

Findings

Exact saturation number for $k$-edge-connected graphs: $(k-1)(n-1)-loor{rac{n}{k+1}}{k-1 race 2}$

Exact extremal number for $k$-edge-connected graphs: $(k-1)n - {k race 2}$

Lower bounds on spectral radius for saturated 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 $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ 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 $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n,\mathcal{F}_k)=(k-1)n-{k\choose2}$, we prove $sat(n,\mathcal{F}'_k)=(k-1)(n-1)-\lfloor{\frac {n}{k+1}}\rfloor{k-1 \choose 2}$. We also prove $ex(n,\mathcal{F}'_k)=(k-1)n-{k\choose2}$ and characterize when equality holds. Finally, we give a lower bound on the spectral radius for $\mathcal{F}_k$-saturated and $\mathcal{F}'_k$-saturated graphs.