Some remarks on Hajós’ conjecture

Hajós’ conjecture is false for almost all graphs but only few explicit counterexamples have appeared in the literature. We relate Hajós’ conjecture to Ramsey theory, perfect graphs, and the maximum cut problem and obtain thereby new classes of explicit counterexamples. On the other hand, we show that some of the graphs which Catlin conjectured to be counterexamples to Hajós’ conjecture satisfy the conjecture, and we characterize completely the graphs which satisfy Catlinʹs conjecture.