Cartesian products of graphs and their coherent configurations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910685480353792 |
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| author | Cai, Jinzhuan Guo, Jin Gavrilyuk, Alexander L. Ponomarenko, Ilia |
| author_facet | Cai, Jinzhuan Guo, Jin Gavrilyuk, Alexander L. Ponomarenko, Ilia |
| contents | 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 $m\ge 6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cartesian products of graphs and their coherent configurations Cai, Jinzhuan Guo, Jin Gavrilyuk, Alexander L. Ponomarenko, Ilia Combinatorics 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 $m\ge 6$. |
| title | Cartesian products of graphs and their coherent configurations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.02689 |