Graph coloring: The more colors, the better?

In this paper, we investigate the algorithmic complexity of deciding colorability, as a function of the number of available colors. Intuitively, one may assume that the problem's complexity is highest around the chromatic number of the graph. We give substantial empirical evidence that this intuition is largely true, both for exact and heuristic graph coloring algorithms. We give a rigorous proof that the complexity of a class of exact algorithms is monotonously increasing in the number of available colors in the non-colorable case, and give a counter-example to demonstrate that the analogous claim does not always hold for colorable graphs.