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We discuss many coloring problems. Most notably, we devise a modified algorithm for 3-coloring that has a worst-case time bound of O(1.3236^n). We obtain this by adapting Beigel and Eppstein's time O(1.3289^n) algorithm. They found a structure in the graph that could be colored relatively easily called a maximal bushy forest, which we modify to limit the number of relatively difficult-to-color vertices by dividing the vertices in the graph using their relation to the bushy forest. Furthermore, we take a look at how a small vertex cover can be used to quickly find a k-coloring of a graph. Finally, we contribute new lower bounds for maximal induced k-colorable subgraphs, discuss the properties of these graphs and the upper bound on the number of maximal induced k-colorable subgraphs on graphs with these properties.
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