Remark: Family 157. Counterexamples to the Hadwiger and Colin de Verdi conjectures

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy ${\chi }_f(G)>h(G)$, where $h(G)$ is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound ${\chi }_f(G)\le \mu (G)+1$. In the positive direction, every finite nonempty graph satisfies ${\chi }_{\mathrm{list}}(G)\le Ch(G)$ for a universal constant $C$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.