Abstract
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 \(k=\varOmega (\sqrt{n})\).
O. Verbitsky was supported by DFG grant KO 1053/8-1. He is on leave from the IAPMM, Lviv, Ukraine.
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Notes
- 1.
A simple inspection of the proof shows that Theorem 7 can be extended to the range of n divisible by 8p for each prime p.
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Fuhlbrück, F., Köbler, J., Ponomarenko, I., Verbitsky, O. (2021). The Weisfeiler-Leman Algorithm and Recognition of Graph Properties. In: Calamoneri, T., Corò, F. (eds) Algorithms and Complexity. CIAC 2021. Lecture Notes in Computer Science(), vol 12701. Springer, Cham. https://doi.org/10.1007/978-3-030-75242-2_17
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