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 $C\cap S\ne \varnothing$ for every $C\in \mathbb{C}$ and $S\in \mathbb{S}$. It has been conjectured by De Simone and Körner 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.

中文翻译：

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证明法线图猜想

如果存在两个覆盖，则图G称为法线，$\mathbb{C}$ 和 $\mathbb{小号}$ 的顶点集 $\mathbb{C}$诱导一个集团ģ，的每一个成员$\mathbb{小号}$诱导一组独立的g ^和$C\cap \mathrm{小号}\ne \varnothing$ 每一个 $C\in \mathbb{C}$ 和 $\mathrm{小号}\in \mathbb{小号}$。它于1999年被臆想由De Simone和克尔纳一个图形摹是正常的，如果摹不含$C_5$， $C_7$ 和 $\stackrel{〜}{C_7}$作为诱导子图。我们反对这一猜想。