4‐Separations in Hajós graphs
Qiqin Xie, Shijie Xie, Xingxing Yu, Xiaofan Yuan · Journal of Graph Theory · 2021
Abstract As a natural extension of the Four Color Theorem, Hajós conjectured that graphs containing no ‐subdivision are 4‐colorable. Any possible counterexample to this conjecture with minimum number of vertices is called a Hajós graph. Previous results show that Hajós graphs are 4‐connected but not 5‐connected. A ‐separation in a graph is a pair of edge‐disjoint subgraphs of such that , , and for . In this paper, we show that Hajós graphs do not admit a 4‐separation such that and can be drawn in the plane with no edge crossings and all vertices in incident with a common face. This is a step in our attempt to reduce Hajós' conjecture to the Four Color Theorem.