A graph $G$ is called normal if there exist two coverings, $\mathbb{ C}$ and $\mathbb{ S}$ of its vertex set such that every member of $\mathbb{ C}$ induces a clique in $G$, every member of $\mathbb{ S}$ induces an independent set in $G$ and $C \cap S \neq \emptyset$ for every $C \in \mathbb{ C}$ and $S \in \mathbb{ S}$. It has been conjectured by De Simone and K\"orner in 1999 that a graph $G$ is normal if $G$ does not contain $C_5$, $C_7$ and $\overline{ C_7}$ as an induced subgraph. We disprove this conjecture.
Categories: math.CO