On a Coloring Conjecture of Hajós

Hajós conjectured that graphs containing no subdivision of $$K_5$$K5 are 4-colorable. It is shown in Yu and Zickfeld (J Comb Theory Ser B 96:482–492, 2006) that if there is a counterexample to this conjecture then any minimum such counterexample must be 4-connected. In this paper, we further show that if $$G$$G is a minimum counterexample to Hajós’ conjecture and $$S$$S is a 4-cut in $$G$$G then $$G-S$$G-S has exactly two components.