Colour-critical vertex
= Colour-critical vertex
{title2=$\chi(F-v)=\chi(F)-1$}
A vertex $v$ is colour-critical if deleting it lowers the <chromatic number> by one. If $\chi(F)=r+1$ and $\chi(F-v)=r$, an embedding of $F-v$ into a complete $r$-partite graph inside one neighbourhood extends to an embedding of $F$. This supplies maximum-degree obstructions in <extremal graph theory>.