The Weisfeiler-Leman Algorithm and Recognition of Graph Properties.

The k-dimensional Weisfeiler-Leman algorithm (\(k\text {-}\mathrm {WL}\)) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of \(k\text {-}\mathrm {WL}\) to recognition of graph properties. Let G be an input graph with n vertices. We show that, if n is prime, then vertex-transitivity of G can be seen in a straightforward way from the output of \(2\text {-}\mathrm {WL}\) on G and on the vertex-individualized copies of G. This is perhaps the first non-trivial example of using the Weisfeiler-Leman algorithm for recognition of a natural graph property rather than for isomorphism testing. On the other hand, we show that, if n is divisible by 16, then \(k\text {-}\mathrm {WL}\) is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with n vertices unless \(k=\varOmega (\sqrt{n})\).