Can any one help me prove - A simple critically 3-chromatic graph without isolated vertices has $\Delta(G)=2$.
I tried to do it by contradiction and show that if a vertex $v$ has degree 3 or more then the graph $G$ is not critically 3-colourable.
Take a vertex $u$ and its neighbours $v_1,v_2,v_3,...,v_n$. Then $G-u$ or $G-v_i$ is not 2 colourable because... I'm not sure to finish it off? Or I could be heading in the wrong direction?