A degree-five vertex is inserted into a plane graph after one explicit Kempe-chain colour swap

Before $v$ is inserted, its five prospective neighbours use all five colours. In the subgraph induced by colours $1$ and $3$, both $v_1$ and $v_3$ are isolated: their two cycle neighbours have colours $2,5$ and $2,4$, respectively. In particular there is no alternating $1$-$3$ path between them, consistently with [L2].

Swap colours $1$ and $3$ on the Kempe component $\{v_1\}$. The colouring remains proper by [L1]; now both $v_1$ and $v_3$ have colour $3$, but they are nonadjacent, and no rim vertex has colour $1$. Give the inserted centre $v$ colour $1$. Every spoke then has differently coloured endpoints, producing an explicit five-colouring of the plane wheel and illustrating the swap used in Five colour theorem: every planar graph has chromatic number at most five.