Karger's algorithm: why does every vertex have degree at least the number of edges crossing a min cut?
You pretty much have the answer in front of you. 

For any vertex $v$, the cut $\left(\left\{v\right\},V\setminus\left\{v\right\}\right)$ has $\text{deg}(v)$ crossing edges. 

If there exists a vertex $v$ with degree $&lt;k$, then its corresponding cut has less than $k$ crossing edges, contradicting the minimality of $(A,B)$.