3-Coloring in Time O(1.3217^n)

We propose a new algorithm for 3-coloring that runs in time O(1.3217^n). For this algorithm, we make use of the time O(1.3289^n) algorithm for 3-coloring by Beigel and Eppstein. They described a structure in all graphs, whose vertices could be colored relatively easily. In this paper, we improve upon this structure and present new ways to determine how the involved vertices reduce the runtime of the algorithm.