Cartesian products of graphs and their coherent configurations

Abstract

The coherent configuration $\mathrm{WL}\left(X\right)$ of a graph X is the smallest coherent configuration on the vertices of X that contains the edge set of X as a relation. The aim of the paper is to study $\mathrm{WL}\left(X\right)$ when X is a Cartesian product of graphs. The example of a Hamming graph shows that, in general, $\mathrm{WL}\left(X\right)$ does not coincide with the tensor product of the coherent configurations of the factors. We prove that if X is “closed” with respect to the 6-dimensional Weisfeiler-Leman algorithm, then $\mathrm{WL}\left(X\right)$ is the tensor product of the coherent configurations of certain graphs related to the prime decomposition of X. This condition is trivially satisfied for almost all graphs. In addition, we prove that the property of a graph “to be decomposable into a Cartesian product of k connected prime graphs” for some $k\ge 1$ is recognized by the m-dimensional Weisfeiler-Leman algorithm for all $m\ge 6$.