# Criticality of a graph

What's the usual method for proving the criticality of a graph?

I've been trying out different methods and theorems but I can't find a decent method that's really convincing.

If $G$ is $n$-critical (i.e. $\chi(G) = n$), then $\delta(G) \geq n-1$. So if $\delta(G) < n-1$ then $G$ is not $n$-critical.
If $G$ is $n$-critical then it is the complete graph. – Yuval Filmus Feb 14 '11 at 2:28