Odd Hadwiger conjecture for graphs with independence number two

The question asks whether every $n$-vertex graph with $\alpha(G)=2$ contains an odd $K_{\lceil n/2\rceil}$ minor. This assertion is now disproved, while a weaker disjunctive replacement has been proved.

Known results

• Kawarabayashi and Song’s $\lceil n/3\rceil$ odd-minor bound is asymptotically best possible for $\alpha(G)\le 2$.
• Chen and Deng proved the $\lceil n/2\rceil$ odd-minor conclusion when the graph is not $\lceil n/2\rceil$-connected.
• Structural hypotheses involving large cliques or forbidden induced subgraphs also imply the $\lceil n/2\rceil$ odd minor.
• Every such graph contains $K_{\ell,\chi(G)-\ell}$ as an odd minor for $1\le \ell<\chi(G)$ (2025).

August 2026 replacement theorem

Bermúdez, Netto, and Quiroz’s preprint, dated August 15, proves that every graph with $\alpha(G)=2$ contains $K_{\lceil n/2\rceil}$ as an odd minor or as a totally odd immersion. It also gives counterexamples to the minor-only assertion.

Current status (as of August 2026): The original odd-minor assertion is disproved for $\alpha(G)=2$, and the replacement odd-minor-or-totally-odd-immersion theorem is stated in a preprint; related minor-only questions remain open.