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The finite $k$-set homogeneous graphs

Cai Heng Li, Fu-Gang Yin, Jin-Xin Zhou

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.

The finite $k$-set homogeneous graphs

Abstract

A classification is given of finite -set-homogeneous graphs for , leading to a striking result that each finite -set-homogeneous graph is -homogeneous. It shows that -set-homogeneous graphs are rare, consisting of the following graphs and their complements: , , , 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 are self-paired, except for acting on 36 points.
Paper Structure (20 sections, 43 theorems, 65 equations, 2 tables)

This paper contains 20 sections, 43 theorems, 65 equations, 2 tables.

Table of Contents

  1. Introduction
  2. A reduction
  3. Primitive permutation groups of rank $5$
  4. Affine primitive permutation groups of rank 4
  5. Type (A1)
  6. Type (A2)
  7. Type (A3)
  8. Types (A4)--(A11)
  9. Class (B)
  10. Class (C)
  11. Almost simple primitive permutation groups of rank 4
  12. Proof of Theorems and
  13. Finite $3$-set-homogeneous graphs
  14. Preliminaries
  15. A reduction
  16. ...and 5 more sections

Key Result

Theorem 1.2

For each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous. Moreover, if ${\it\Gamma}$ is a $k$-homogeneous graph with $k\geqslant 2$, then one of the following holds:

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • ...and 43 more