3-colorability of graphs with minimum degree at least 6

Let $G$ be an $n$-vertex graph and let $L:V(G)\to P(\{1,2,3\})$ 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 $i\in {2,3}$\mathrm{.}Asacorollary,weconcludethatthe3-colorabilityofanygraph$G$withminimumdegreeatleast6canbedeterminedin$O(1.3196^{n-.5\Delta(G)})$ time.