A characterization of graphs with no $K_{3,4}$ minor

TL;DR

This paper provides a complete structural characterization of graphs excluding the $K_{3,4}$ minor, with implications for connectivity, Hamiltonian properties, and embeddings.

Contribution

It offers a comprehensive characterization of $K_{3,4}$-minor-free graphs and proves new results on their connectivity, Hamiltonian connectivity, and torus embeddings.

Findings

4-connected non-planar graphs with ≥7 vertices and min degree ≥5 contain $K_{3,4}$ and $K_6^-$ minors

Every 4-connected $K_{3,4}$-minor-free graph is Hamiltonian-connected

Such graphs can be embedded on the torus

Abstract

A complete structural characterization of graphs with no $K_{3,4}$ minor is obtained, and the following consequences are established. Every $4$-connected non-planar graph with at least seven vertices and minimum degree at least five contains both $K_{3,4}$ and $K_6^-$ as minors, thereby proving a conjecture of Kawarabayashi and Maharry in a strengthened form. Moreover, every $4$-connected graph with no $K_{3,4}$ minor is hamiltonian-connected, extending a theorem of Thomassen, and admits an embedding on the torus.