  Let $G$ be an $n$-vertex graph and let $L:V(G)\rightarrow P(\{1,2,3\})$ be a
list assignment over the vertices of $G$, where each vertex with list of size 3
and of degree at most 5 has at least three neighbors with lists of size 2. We
can determine $L$-choosability of $G$ in $O(1.3196^{n_3+.5n_2})$ time, where
$n_i$ is the number of vertices in $G$ with list of size $i$ for $i\in
\{2,3\}$. As a corollary, we conclude that the 3-colorability of any graph $G$
with minimum degree at least 6 can be determined in $O(1.3196^{n-.5\Delta(G)})$
time.
