Wheels in planar graphs and Haj\'os graphs

TL;DR

This paper investigates Hajós graphs, which are potential minimal counterexamples to Hajós' conjecture, showing that certain cuts with planar sides impose size or wheel structure constraints, aiding in reducing the conjecture to the Four Color Theorem.

Contribution

It establishes structural properties of Hajós graphs with specific cuts, advancing the approach to prove Hajós' conjecture via the Four Color Theorem.

Findings

Hajós graphs with 4-cuts or 5-cuts with planar sides have small or wheel-containing sides.

The results restrict the structure of potential counterexamples to Hajós' conjecture.

These structural insights help in reducing Hajós' conjecture to the Four Color Theorem.

Abstract

It was conjectured by Haj\'{o}s that graphs containing no $K_5$-subdivision are 4-colorable. Previous results show that any possible minimum counterexample to Haj\'{o}s' conjecture, called Haj\'{o}s graph, is 4-connected but not 5-connected. In this paper, we show that if a Haj\'{o}s graph admits a 4-cut or 5-cut with a planar side then the planar side must be small or contains a special wheel. This is a step in our effort to reduce Haj\'{o}s' conjecture to the Four Color Theorem.