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arXiv · 2602.19882

The finite $k$-set homogeneous graphs

Cai Heng Li·Fu-Gang Yin·Jin-Xin Zhou
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Abstract

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.

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Cai Heng Li, Fu-Gang Yin, Jin-Xin Zhou. 2026-02-23. The finite $k$-set homogeneous graphs. https://arxiv.org/abs/2602.19882

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