The complixity of McKay's canonical labeling algorithm

We study the time complexity of McKay's algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered auto-morphisms and by hashing partial information of vertex labelings. In practice, the algorithm is implemented in the nauty package. We obtain colorings of F urer's graphs that allow the algorithm to compute their canonical forms in polynomial time. We then prove an exponential lower bound of the algorithm for connected 3-regular graphs of color-class size 4 using F urer's construction. We conducted experiments with nauty for these graphs. Our experimental results also indicate the same exponential lower bound.