The Colin de Verdiere Chromatic Conjecture

Theorem 1.1: there are graphs of arbitrarily large order with $\alpha(G)\le2$ and $\mu(G)+1<|V(G)|/2\le\chi(G)$; the fractional chromatic number also exceeds $\mu(G)+1$. The proof shows every well-signed matrix on the graph with one negative eigenvalue has rank above $|V(G)|/2+1$, without using the Strong Arnold Property, so the Lovasz-Schrijver parameter $\kappa$ also satisfies $\kappa(G)+1<|V(G)|/2$. Since $h\le\mu+1$, these graphs also violate Hadwiger's inequality, by a route independent of the connected-matching companion. Nothing is claimed for small $\mu$.

Theorem 1.1: there are graphs of arbitrarily large order with $\alpha(G)\le2$ and $\mu(G)+1<|V(G)|/2\le\chi(G)$; the fractional chromatic number also exceeds $\mu(G)+1$. The proof shows every well-signed matrix on the graph with one negative eigenvalue has rank above $|V(G)|/2+1$, without using the Strong Arnold Property, so the Lovasz-Schrijver parameter $\kappa$ also satisfies $\kappa(G)+1<|V(G)|/2$. Since $h\le\mu+1$, these graphs also violate Hadwiger's inequality, by a route independent of the connected-matching companion. Nothing is claimed for small $\mu$.