The complexity 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 algo rithm is based on a type of backtrack search, and it performs pruning by disc overed automorphisms and by hashing partial information of vertex labelin gs. In practice, the algorithm is implemented in the nautypackage. We obtain colorings of Fürer’s graphs that allow the algorithm to compute their canonical f orms in polynomial time. We then prove an exponential lower bound of the algorit hm for connected 3-regular graphs of color-class size 4 using Fürer’s construction. We conducted experiments withnautyfor these graphs. Our experimental results also indicate the same exponential lower bound.