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

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.

💡 Research Summary

The paper tackles the long‑standing problem of characterising all finite vertex‑transitive connected graphs whose connectivity is less than five. Building on earlier work that settled the cases of connectivity ≤ 2 (Droms, Servatius, Servatius) and connectivity = 3 (Carmesin and Kurkofka), the authors extend the classification to connectivity = 4. The central methodological innovation is a Tutte‑type canonical decomposition that works for 4‑connected graphs, called the “tetra‑decomposition”. This decomposition uniquely splits any 2‑connected graph along 2‑separators into pieces that are either 3‑connected, cycles, or K₂, and records how these pieces are glued together in a decomposition graph that may be a tree or a cycle (the latter leading to “cycle‑decompositions”).

The authors introduce several refined connectivity notions. A graph is quasi‑4‑connected if it is 3‑connected and every 3‑separator leaves a side consisting of a single vertex. A graph is quasi‑5‑connected if every 4‑separator leaves a side of size at most two. A 2‑quasi‑5‑connected graph is quasi‑4‑connected and satisfies the latter condition. They also define H‑expansions (replacing each vertex of a regular graph G by a copy of H and matching edges accordingly) and focus on the special cases of K₄‑expansions and C₄‑expansions.

The first major result (Theorem 1.2, originally from