Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

On-Hei Solomon Lo

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.

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

Abstract

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

Paper Structure

This paper contains 19 sections, 48 theorems, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Splitter theorems
  4. Seymour's splitter theorem
  5. Internally $4$-connected graphs
  6. Non-embeddable extensions
  7. $4$-connected graphs with no $K_{3,4}$ minor
  8. The projective-planar case
  9. The non-projective-planar case
  10. Three Archdeacon graphs and oloidal graphs
  11. Containing a spanning subdivision of $E_{20}$
  12. Containing a spanning subdivision of $F_4$
  13. Proof of Theorem \ref{['thm:4cnppK34mf']}
  14. Proof of Theorem \ref{['thm:4cK34mf']}
  15. Generating all graphs with no $K_{3,4}$ minor
  16. ...and 4 more sections

Key Result

Theorem 1.2

A graph $G$ is $4$-connected and has no $K_{3,4}$ minor if and only if one of the following holds:

Figures (9)

  • Figure 1: The graphs $K_{3,4}$ (left) and $K_6^-$ (right).
  • Figure 2: The $H$-piece (left), $Y$-piece (middle), and $I$-piece (right). Applying the corresponding patching adds two, one, or no new patches, respectively, indicated by the shaded regions.
  • Figure 3: The initial patch construct $(G_0,\mathcal{P}_0)$ embedded in the projective plane. The outer circle represents the cross-cap, and the shaded region is the unique face of length $4$.
  • Figure 4: The internally $4$-connected non-projective-planar graphs $D_{17}$ (left), $E_{20}$ (middle), and $F_4$ (right).
  • Figure 5: The $4$-connected non-projective-planar graphs $\mathfrak{D}_{s^1,s^2}$ (left), $\mathfrak{E}_{s}$ (middle), and $\mathfrak{F}$ (right).
  • ...and 4 more figures

Theorems & Definitions (81)

  • Conjecture 1.1: Kawarabayashi2012
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 3.1: Seymour1980Negami1982
  • Proposition 3.2: Johnson2002
  • Lemma 3.3: Johnson2002
  • Theorem 3.4: Hegde2018
  • Theorem 4.1: Maharry2012
  • ...and 71 more