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On the dichromatic number in kernel theory. (English) Zbl 0937.05048
Given a digraph $$D$$, a subset $$N$$ of its vertex set $$V(D)$$ is called a kernel of $$D$$ if it is independent (there is no arc joining two vertices of $$N$$) and absorbing (for each $$x\notin N$$ there is an arc $$xy$$ with $$y\in N$$). If every induced subdigraph of $$D$$ has a kernel, then $$D$$ is said to be kernel-perfect. The dichromatic number of $$D$$ is the minimum number of acyclic classes into which $$V(D)$$ can be partitioned. The authors give constructions of kernel-perfect digraphs with a prescribed dichromatic number and without semicycles of lengths 2 or 3. Strengthening their previous results in [Discrete Math. 94, No. 3, 181-187 (1991; Zbl 0748.05060)] they provide also results and conjectures concerning critical kernel-imperfect digraphs.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C15 Coloring of graphs and hypergraphs
##### Keywords:
digraph; dichromatic number; kernel; kernel-perfect
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##### References:
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