minimum degree of 2-connected graph

Question

If we have a 2-connected graph G, then can we say that δ(G) > k(G)? I need that in order to use Halin's theorem. Can we have then a more general relation between k-connected graphs and their minimum degree?

Answer 1

For every simple graph $G$: $$\delta(G)\geq\kappa(G).$$

Here's why: Take the vertex $v \in V(G)$ with $\deg(v)=\delta(G)$. Take a set $S=\{u \in V(G) | uv \in E(G)\}$. Obviously $G-S$ is disconnected. Therefore: $$\kappa(G)\leq |S|=\delta(G).$$

So to answer your question, $\delta(G)$ and $\kappa(G)$ can also be equal.