Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree

TL;DR

The paper investigates the motion $\mu(\Gamma)$ of vertex-transitive graphs by linking graph automorphism groups to the structure of transitive permutation groups with small minimal degree. It develops a wreath-product framework and conducts a case analysis for minimal degree $p$ (prime) and for minimal degree $4$, leveraging Jones (2014) for primitive groups and providing CFSG-free graph consequences where possible. It then delivers a near-complete classification of vertex-transitive graphs with $\mu(\Gamma)=2$ or $\mu(\Gamma)=4$, showing that no graphs have motion equal to an odd prime $p\ge3$, except in specific wreath-product forms, and introducing the Inf construction to realize additional $\mu(\Gamma)=4$ examples such as $C_5\wr \Theta$, $ (K_m\square K_2)\wr \Theta$, and their complements. These results deepen understanding of graph symmetry, yield explicit construction templates, and connect automorphism-group theory with concrete graph-structural descriptions, with some results relying on CFSG in the group-theoretic parts but offering CFSG-free graph conclusions.

Abstract

The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper we develop some results on permutation groups (primitive and imprimitive) with small minimal degree. As a consequence of such results we classify vertex-transitive graphs whose motion is $4$ or a prime number.

Figures (3)

• Figure 1: Local view of the graph $\Inf$ with $m=4$.
• Figure 2:
• Figure 3:

Theorems & Definitions (56)

• Theorem 1.1
• Corollary 1.2
• Definition 1.3
• Remark 1.4
• Example 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Theorem 1.10