We study several extensions of the notion of perfect graphs to k-uniform hypergraphs. One main definition extends to hypergraphs the notion of perfect graphs based on coloring. Let G be a k-uniform hypergraph. A coloring of a k-uniform hypergraph G is proper if it is a coloring of the (k − 1)-tuples with elements in V(G) in such a way that no edge of G is a monochromatic \(K_k^{k - 1}\).

A k-uniform hypergraph G is C_ω-perfect if for every induced subhypergraph G′ of G we have:

• if X ⊆ V(G′) with ∣X∣ < k − 1, then there is a proper (ω(G′) − k + 2)-coloring of G′ (so (k − 1)-tuples are colored) that restricts to a proper (ω(G′) − k + 2)-coloring of lk_G′(X) (so (k − ∣X∣ − 1)-tuples are colored).

Another main definition is the following: A k-uniform hypergraph G is hereditary perfect (or, briefly, H-perfect) if all links of sets of (k − 2) vertices are perfect graphs.

The notion of C_ω perfectness is not closed under complementation (for k > 2) and we define G to be doubly perfect if both G and its complement are C_ω perfect. We study doubly-perfect hypergraphs: In addition to perfect graphs nontrivial doubly-perfect graphs consist of a restricted interesting class of 3-uniform hypergraphs, and within this class we give a complete characterization of doubly-perfect H-perfect hypergraphs.

我们研究了完美图概念到k均匀超图的几种扩展。超图的一个主要定义是基于着色的完美图的概念。令G为k一致超图。如果k -均匀超图G的着色是具有V ( G )中元素的 ( k − 1)-元组的着色，并且G的任何边都不是单色的\(K_k^{k - 1}\) .

如果对于G的每个导出子超 图G ′我们_有：

• 如果X ⊆ V ( G ′) 且 ∣ X ∣ < k − 1，则G ′存在适当的 ( ω ( G ′) − k + 2)-着色（因此 ( k − 1)-元组被着色）限制为lk _G ′ ( X ) 的正确 ( ω ( G _′ ) − k + 2) 着色（因此 ( k − ∣ X ∣ − 1) 元组被着色）。

另一个主要定义如下：如果 ( k − 2)个顶点集的所有链接都是完美图，则k一致超图G是遗传完美的（或者简单地说，H完美的）。

C _ω完美性的概念在互补条件下并不封闭（对于k > 2），如果G及其补体都是C _ω完美性，则我们将G定义为双重完美性。我们研究双完美超图：除了完美图之外，非平凡的双完美图还包含一类受限制的有趣的 3-一致超图，在这个类中，我们给出了双完美 H 完美超图的完整表征。