Complexity of coloring random graphs: zooming in on the hardest part∗

It is known that the problem of deciding k-colorability of a graph exhibits an easy-hard-easy pattern, with the maximal complexity being either at k = χ−1 or k = χ , where χ is the chromatic number of the graph. However, the behavior around the complexity peak is poorly understood. In this paper, we use list coloring to model coloring with a fractional number of colors between χ − 1 and χ . We present a comprehensive computational study on the complexity of graph coloring in this critical range. According to our findings, an easy-hard-easy pattern can be observed on a finer scale between χ−1 and χ as well. The highest complexity found this way can be higher than for any integer value of k. It turns out that the complexity follows a periodic 3-dimensional pattern; understanding these patterns is very important for benchmarking purposes. Our results also answer the previously open question whether coloring with χ−1 or with χ colors is harder: this depends on the location of the maximal fractional complexity.