On the Weisfeiler-Leman dimension of circulant graphs

TL;DR

This paper investigates the Weisfeiler-Leman dimension of circulant graphs, establishing an upper bound related to the number of prime divisors of the graph's order, which aids in graph isomorphism testing.

Contribution

It provides a new upper bound on the Weisfeiler-Leman dimension for circulant graphs based on their order's prime factorization.

Findings

Weisfeiler-Leman dimension is at most Ω(n)+3 for circulant graphs of order n.

The bound relates the dimension to the number of prime divisors of n.

This result advances understanding of graph isomorphism testing for circulant graphs.

Abstract

A circulant graph is a Cayley graph of a finite cyclic group. The Weisfeiler-Leman-dimension of a circulant graph $X$ with respect to the class of all circulant graphs is the smallest positive integer~$m$ such that the $m$-dimensional Weisfeiler-Leman algorithm correctly tests the isomorphism between $X$ and any other circulant graph. It is proved that for a circulant graph of order $n$ this dimension is less than or equal to $\Omega(n)+3$, where $\Omega(n)$ is the number of prime divisors of~$n$.