The Weak Graph Complement Conjectures

Let $G$ be a graph. Write $G^c$ for its complement, $\operatorname{mr}(G)$ for minimum rank, $\operatorname{mr}_+(G)$ for positive semidefinite minimum rank, $\nu(G)$ for the positive semidefinite Colin de Verdière parameter, $\operatorname{mr}_\nu(G)=\ord G-\nu(G)$, and $\ord G$ for the number of vertices of $G$.

Weak Graph Complement Conjectures. There exist universal constants $b,b_+,b_\nu<2$ such that, for every graph $G$,

$$\operatorname{mr}(G)+\operatorname{mr}(G^c)\leq b\cdot\ord G+2,$$ $$\operatorname{mr}_+(G)+\operatorname{mr}_+(G^c)\leq b_+\cdot\ord G+2,$$

and

$$\operatorname{mr}_\nu(G)+\operatorname{mr}_\nu(G^c)\leq b_\nu\cdot\ord G+2.$$

These are weaker asymptotic forms of the three graph-complement conjectures. The source later states that the strongest version is resolved with constants at most $1.708$, but does not provide a general exact optimal constant in the supplied material.