The Weisfeiler-Leman algorithm and recognition of graph properties

The k-dimensional Weisfeiler-Leman algorithm (k-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of k-WL to recognition properties. Let G be an input with n vertices. show that, if prime, then vertex-transitivity can seen straightforward way from output 2-WL on and vertex-individualized copies G. This perhaps first non-trivial example using for natural property rather than On other hand, we divisible by 16, unable distinguish between vertex-transitive non-vertex-transitive graphs vertices unless k=Ω(n). Similar results are obtained arc-transitivity. Our lower bounds based analysis Cai-Fürer-Immerman construction, which might independent interest. In particular, provide sufficient conditions under made colorless.