Towards faster isomorphism tests for bounded-degree graphs

Luks’s algorithm (JCSS 1982) to test isomorphism of bounded degree graphs in polynomial time is one of the most important results in the context of the Graph Isomorphism Problem and has been repeatedly used as a basic building block for many other algorithms. In particular, for graphs of logarithmic degree, Babai’s quasipolynomial isomorphism test (STOC 2016) essentially boils down to Luks’s algorithm, and any improvement of Babai’s algorithm requires an improved isomorphism test for graphs of (poly)logarithmic degree. In this work, we obtain such an improvement: we give an algorithm that solves the Graph Isomorphism Problem in time n d) c), where n is the number of vertices of the input graphs, d is the maximum degree of the input graphs, and c is an absolute constant. The best previous isomorphism test for graphs of maximum degree d due to Babai, Kantor and Luks (FOCS 1983) runs in time n log . Our result generalizes the quasipolynomial-time algorithm for the general isomorphism problem due to Babai.