Definitions

# K-edge-connected graph

In graph theory, a graph $G$ with edge set $E\left(G\right)$ is said to be $k$-edge-connected if $G setminus X$ is connected for all $X subseteq E\left(G\right)$ with $left| X right| < k$. In plain English, a graph is $k$-edge-connected if the graph remains connected when you delete fewer than $k$ edges from the graph.

If a graph $G$ is $k$-edge-connected then $k le delta\left(G\right)$, where $delta\left(G\right)$ is the minimum degree of any vertex $v in V\left(G\right)$. This fact is clear since deleting all edges with an end at a vertex of minimum degree will disconnect that vertex from the graph.