Hajós’s subdivision conjecture for t equal to five and six

Determine whether every graph with chromatic number at least t contains a subdivision of K_t for t in {5,6}.

Background

The Hajós conjecture is a strengthening of Hadwiger’s conjecture that replaces the existence of a K_t minor with the existence of a subdivision of K_t. The paper states that Catlin disproved the conjecture for all t at least 7, while the cases t=5 and t=6 remain unresolved.

References

This stronger conjecture was refuted by Catlin in 1979 for all $t \ge 7$ and remains open for $t\in \set{5,6}$.

— Disproof of the dominating Hadwiger conjecture  (2609.35361 - Illingworth et al., 28 Sep 2026) in Section 1, Introduction