Wheels in planar graphs and Hajós graphs
Qiqin Xie, Shijie Xie, Xingxing Yu, Xiaofan Yuan · Journal of Graph Theory · 2021
Abstract It was conjectured by Hajós that graphs containing no ‐subdivision are 4‐colorable. Previous results show that any possible minimum counterexample to Hajós' conjecture, called Hajós graph, is 4‐connected but not 5‐connected. In this paper, we show that if a Hajós 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 Hajós' conjecture to the Four Color Theorem.