The finite $k$-set homogeneous graphs

TL;DR

This paper classifies finite k-set-homogeneous graphs for k≥2, revealing they are also k-homogeneous, and identifies the rare 3-set-homogeneous graphs along with their complements, contributing to graph symmetry understanding.

Contribution

It provides a complete classification of finite k-set-homogeneous graphs for k≥2 and shows their equivalence to k-homogeneity, also identifying all 3-set-homogeneous graphs and their complements.

Findings

Finite k-set-homogeneous graphs are k-homogeneous.

3-set-homogeneous graphs are rare and explicitly classified.

All orbitals in a primitive permutation group of rank 4 are self-paired, except one case.

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 Schl\"{a}fli 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.