The bipartite-complement conjecture for extremal edge polytopes

Let $G$ be a finite simple graph in $\Omega_d$ attaining the maximum number of edges of its edge polytope, so that $\mu_d=\varepsilon(G)$.

Bipartite-complement conjecture. The complement of $G$ is a bipartite graph.

This conjecture proposes a structural description of graphs maximizing the number of edges of an edge polytope. The preceding theorem shows that complete graphs are extremal for $3\leq d\leq13$, while for $d\geq15$ some graphs have more edge-polytope edges than the complete graph; determining the exact maximum and an extremal graph remains unsolved for $d\geq15$.