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Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be afamily of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring$\phi$ of $G$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $G$.Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$,$|L(v)|\ge3$ for every $v\in V(C)-\{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\inV(G)-V(C)$, then $G$ has an $L$-coloring. What happens when $v_1$ and $v_2$ arenot adjacent? Then an $L$-coloring need not exist, but in the first paper ofthis series we have shown that it exists if $|L(v_1)|,|L(v_2)|\ge2$. Here wecharacterize when an $L$-coloring exists if $|L(v_1)|\ge1$ and $|L(v_2)|\ge2$.  This result is a lemma toward a more general theorem along the same lines,which we will use to prove that minimally non-$L$-colorable planar graphs withtwo precolored cycles of bounded length are of bounded size. The latter resulthas a number of applications which we pursue elsewhere.

Five-list-coloring graphs on surfaces III. One list of size one and one list of size two