Wheels in planar graphs and Haj\'os graphs
Combinatorics
2019-11-26 v1
Abstract
It was conjectured by Haj\'{o}s that graphs containing no -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.
Cite
@article{arxiv.1911.10464,
title = {Wheels in planar graphs and Haj\'os graphs},
author = {Qiqin Xie and Shijie Xie and Xingxing Yu and Xiaofan Yuan},
journal= {arXiv preprint arXiv:1911.10464},
year = {2019}
}