3-colorability of graphs with minimum degree at least 6

Let G be an n-vertex graph and let L : V (G) → 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(1.3196n3+.5n2) time, where ni is the number of vertices in G with list of size i for i ∈ {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(1.3196n−.5∆(G)) time.