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A characterisation of all vertex-transitive finite graphs of connectivity < 5

Jan Kurkofka, Tim Planken

TL;DR

We classify all vertex-transitive finite connected graphs with connectivity $<5$ by combining Tutte-type canonical decompositions with canonical $Y$--$\Delta$ refinements to handle quasi-$4$-connected cases. The approach yields a unified structural description: graphs are either essentially $5$-connected or belong to explicit families built from cycle-decompositions and $H$-expansions, including the line graph of the cube and its $K_4$- or $C_4$-expansions, with careful treatment of $3$-regular and arc-transitive subcases. The results provide an explicit, canonical classification that ties vertex-transitivity, arc-transitivity, and cycle/bag decompositions into a single framework, enabling direct recognition and structural understanding. This framework extends classical decompositions to the quasi-$4$- and $3$-regular worlds, delivering a comprehensive picture of highly symmetric graphs with low connectivity and establishing concrete, verifiable graph families involved in the classification.

Abstract

We characterise all vertex-transitive finite connected graphs as essentially 5-connected or on a short list of explicit graph-classes. Our proof heavily uses Tutte-type canonical decompositions.

A characterisation of all vertex-transitive finite graphs of connectivity < 5

TL;DR

We classify all vertex-transitive finite connected graphs with connectivity by combining Tutte-type canonical decompositions with canonical -- refinements to handle quasi--connected cases. The approach yields a unified structural description: graphs are either essentially -connected or belong to explicit families built from cycle-decompositions and -expansions, including the line graph of the cube and its - or -expansions, with careful treatment of -regular and arc-transitive subcases. The results provide an explicit, canonical classification that ties vertex-transitivity, arc-transitivity, and cycle/bag decompositions into a single framework, enabling direct recognition and structural understanding. This framework extends classical decompositions to the quasi-- and -regular worlds, delivering a comprehensive picture of highly symmetric graphs with low connectivity and establishing concrete, verifiable graph families involved in the classification.

Abstract

We characterise all vertex-transitive finite connected graphs as essentially 5-connected or on a short list of explicit graph-classes. Our proof heavily uses Tutte-type canonical decompositions.
Paper Structure (5 sections, 46 theorems, 8 equations, 7 figures)

This paper contains 5 sections, 46 theorems, 8 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Analysing 4-connectivity
  3. Analysing 3-regular quasi-4-connectivity
  4. Analysing 3-regular arc-transitivity
  5. Appendix: Analysing 3-connectivity

Key Result

Theorem 1.2

TridecompWe slightly strengthen Tridecomp in the present paper to obtain intro:triVxCon. The use of $H$-expansions is new. A finite connected graph $G$ is vertex-transitive if and only if $G$ is either

Figures (7)

  • Figure 1.1: A cycle of $K_4$-bags
  • Figure 1.2: A cycle alternating between $K_4$-bags and $C_4$-bags
  • Figure 1.3: A cycle alternating between $K_{2,2}$-bags and $C_4$-torsos
  • Figure 1.4: Cycles of triangle-bags
  • Figure 1.5: Cycles of $C_4$-bags
  • ...and 2 more figures

Theorems & Definitions (93)

  • Theorem 1.2
  • Theorem 1
  • Corollary 2
  • Theorem 2.1
  • Corollary 2.2
  • Definition 2.3
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Lemma 2.7
  • ...and 83 more