The finite $k$-set homogeneous graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908848731717632 |
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| author | Li, Cai Heng Yin, Fu-Gang Zhou, Jin-Xin |
| author_facet | Li, Cai Heng Yin, Fu-Gang Zhou, Jin-Xin |
| contents | A classification is given of finite $k$-set-homogeneous graphs for $k\geqslant 2$, leading to a striking result that each finite $k$-set-homogeneous graph is $k$-homogeneous. It shows that $3$-set-homogeneous graphs are rare, consisting of the following graphs and their complements: $\C_5$, $\K_n\square\K_n$, $n\K_m$, the Schläfli graph of order 27, the Higman-Sims graph, the MaLaughlin graph, {affine polar graphs, and elliptic orthogonal graphs}. As an ingredient for the proof, it is shown that all orbitals in a primitive permutation group of rank $4$ are self-paired, except for $\PSU_3(3)$ acting on 36 points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19882 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The finite $k$-set homogeneous graphs Li, Cai Heng Yin, Fu-Gang Zhou, Jin-Xin Group Theory 05C25, 20B25 A classification is given of finite $k$-set-homogeneous graphs for $k\geqslant 2$, leading to a striking result that each finite $k$-set-homogeneous graph is $k$-homogeneous. It shows that $3$-set-homogeneous graphs are rare, consisting of the following graphs and their complements: $\C_5$, $\K_n\square\K_n$, $n\K_m$, the Schläfli graph of order 27, the Higman-Sims graph, the MaLaughlin graph, {affine polar graphs, and elliptic orthogonal graphs}. As an ingredient for the proof, it is shown that all orbitals in a primitive permutation group of rank $4$ are self-paired, except for $\PSU_3(3)$ acting on 36 points. |
| title | The finite $k$-set homogeneous graphs |
| topic | Group Theory 05C25, 20B25 |
| url | https://arxiv.org/abs/2602.19882 |