 Greedy colouring with two precoloured neighbours

Let $x$ have two nonadjacent neighbours $y,z$ in a finite graph $G$, with $G-\{y,z\}$ connected. Then $\chi (G)\le \Delta (G)$. Give $y,z$ one common colour. In a spanning tree of the remaining graph rooted at $x$, order vertices from larger to smaller distance from the root and colour greedily in that order. Every vertex before $x$ has an uncoloured parent, so at most $\Delta (G)-1$ colours are forbidden there. At $x$, the two initial neighbours use the same colour, so again at most $\Delta (G)-1$ distinct colours are forbidden. This constructs a proper colouring from a palette of $\Delta (G)$ colours. It is a useful local step in Brooks' theorem.