The Weisfeiler-Leman Algorithm and Recognition of Graph Properties
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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 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-WL on G and on the vertex-individualized copies of G. However, if n is divisible by 16, then k-WL is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with n vertices as long as k=o(√(n)). Similar results are obtained for recognition of arc-transitivity.
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