Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2011 / A / 2 / a / ii

Write $\Delta =\Delta (G)$, which is at least two because $x$ has the two specified neighbours. Give $y$ and $z$ the same colour, say colour one; this is proper because they are nonadjacent.

Apply part (i) to the connected graph $G\setminus \{y,z\}$, with $x$ last. Colour its vertices greedily in the resulting order using a palette of $\Delta$ colours. Every vertex before $x$ has a neighbour later in this order, so at most $\mathrm{deg}(v)-1\le \Delta -1$ of its neighbours have already received colours. At least one palette colour is available.

When $x$ is reached, all its neighbours are coloured, but the two neighbours $y,z$ have the same colour. Thus there are at most $\mathrm{deg}(x)-1\le \Delta -1$ distinct forbidden colours. Colour $x$ with a remaining one. This constructs a proper colouring withThe common colour on the two initial vertices replaces the usual need for an uncoloured neighbour at the last greedy step.

This is greedy colouring with two precoloured neighbours.