Hadwiger's conjecture

Prove that every graph G contains the complete graph Kχ(G) as a minor, where χ(G) denotes the chromatic number of G.

Background

The paper situates its study of complete minors in the broader problem of determining the largest complete minor from graph parameters such as chromatic number and average degree. It identifies Hadwiger’s conjecture as the central open problem in this area. The conjecture would imply that chromatic number is characterized by the existence of a complete minor of the same order, substantially strengthening known general lower bounds for clique minors.

References

The most important open problem in this direction is Hadwiger's conjecture . This conjecture states that $G$ contains $K_{\chi(G)}$ as a minor, where $\chi(G)$ denotes the chromatic number of $G$.

— Minors in small-set expanders  (2503.06826 - Krivelevich et al., 10 Mar 2025) in Section 1, paragraph “Complete minors”

All the cases $t \ge 7$ of the conjecture remain open.

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

Hadwiger's conjecture asserts that every graph of chromatic number at least $r$ has a $K_r$ minor. It is known for $r\le6$, and a coloring theorem of Delcourt and Postle gives a $K_r$ minor with $r=\Omega(c/\log\log c)$ in every graph of chromatic number $c$.

— Superlinear Quantum Query Lower Bounds for Subgraph Detection  (2609.40263 - Gilani et al., 30 Sep 2026) in Section 1, subsection “Results,” paragraph “Chromatic number”; also Section 1, subsection “Open Problems,” paragraph “Comparison with the chromatic clique”