Extensions of Thomassen's Theorem to Paths of Length At Most Four: Part III

Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. This is the final paper in a sequence of three papers in which we prove some results about partial $L$-colorings $\phi$ of $C$ with the property that any extension of $\phi$ to an $L$-coloring of $\text{dom}(\phi )\cup V(P)$ extends to $L$-color all of $G$, and, in particular, some useful results about the special case in which $\text{dom}(\phi )$ consists only of the endpoints of $P$. We also prove some results about the other special case in which $\phi$ is allowed to color some vertices of $C\setminus \mathring{P}$ but we avoid taking too many colors away from the leftover vertices of $\mathring{P}\setminus \text{dom}(\phi )$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.