  A {\it list assignment} $L$ of a graph $G$ is a function that assigns a set
(list) $L(v)$ of colors to every vertex $v$ of $G$. Graph $G$ is called {\it
$L$-list colorable} if it admits a vertex coloring $\phi$ such that $\phi(v)\in
L(v)$ for all $v\in V(G)$ and $\phi(v)\not=\phi(w)$ for all $vw\in E(G)$.
  The following question was raised by Bruce Richter. Let $G$ be a planar,
3-connected graph that is not a complete graph. Denoting by $d(v)$ the degree
of vertex $v$, is $G$ $L$-list colorable for every list assignment $L$ with
$|L(v)|=\min \{d(v), 6\}$ for all $v\in V(G)$?
  More generally, we ask for which pairs $(r,k)$ the following question has an
affirmative answer. Let $r$ and $k$ be integers and let $G$ be a
$K_5$-minor-free $r$-connected graph that is not a Gallai tree (i.e., at least
one block of $G$ is neither a complete graph nor an odd cycle). Is $G$ $L$-list
colorable for every list assignment $L$ with $|L(v)|=\min\{d(v),k\}$ for all
$v\in V(G)$?
  We investigate this question by considering the components of $G[S_k]$, where
$S_k:=\{v\in V(G) | d(v)<k\}$ is the set of vertices with small degree in $G$.
We are especially interested in the minimum distance $d(S_k)$ in $G$ between
the components of $G[S_k]$.
