Cartesian products of graphs and their coherent configurations

TL;DR

This paper investigates the structure of the coherent configuration of Cartesian product graphs, showing under certain conditions it decomposes into tensor products, and relates graph decomposability to the Weisfeiler-Leman algorithm's dimension.

Contribution

It establishes conditions under which the coherent configuration of a Cartesian product graph decomposes into tensor products, linking graph decomposability to the Weisfeiler-Leman algorithm.

Findings

$ ext{WL}(X)$ does not always match the tensor product of factors.

If $X$ is closed under the 6-dimensional WL algorithm, $ ext{WL}(X)$ is a tensor product.

Graph decomposability is recognized by the $m$-dimensional WL algorithm for $m geq 6$.

Abstract

The coherent configuration $\mathsf{WL}(X)$ 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 $\mathsf{WL}(X)$ when $X$ is a Cartesian product of graphs. The example of a Hamming graph shows that, in general, $\mathsf{WL}(X)$ 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 $\mathsf{WL}(X)$ 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 $m\ge 6$.