The Normal Graph Conjecture

Let $G$ be a graph. A graph is normal if its vertex set has coverings by cliques and by independent sets such that every clique in the first covering intersects every independent set in the second. Let $C_k$ denote the cycle on $k$ vertices, and let $\overline{C_7}$ denote its complement. The Normal Graph Conjecture. If $G$ has no $C_5$, $C_7$ or $\overline{C_7}$ as an induced subgraph, then $G$ is normal. The conjecture was posed by De Simone and Körner in 1999; the source paper is titled “Disproving the normal graph conjecture,” so the claim is refuted.