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Vertex critical r-dichromatic tournaments. (English) Zbl 0532.05031
The dichromatic number d(D) of a directed graph D is the smallest number of colours needed to colour the vertices of D in such a way that D contains no monochromatic directed cycle. If $$d(D)=r$$ but $$d(D-v)=r-1$$ for every vertex v of D, then D is a critical r-dichromatic digraph. The authors construct an infinite family of critical r-dichromatic regular tournaments for every r such that $$r=3$$ or $$r\geq 5$$.
Reviewer: J.W.Moon

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C20 Directed graphs (digraphs), tournaments
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##### References:
 [1] Erdös, P, Problems and results in number theory and graph theory, (), 3-21 [2] P. Erdös and V. Neumann-Lara, On the dichromatic number of a graph, in preparation. [3] Hamidoune, Y.O, On the decomposition of a minimally strongly h-connected digraph into h + 1 acircuitic subgraphs, Discrete math., 31, 89-90, (1980) · Zbl 0437.05027 [4] H. Mayniel, Extension du nombre chromatique et du nombre de stabilité, Preprint. [5] Neumann-Lara, V, The dichromatic number of a digraph, (1981), Instituto de Matemáticas, Universidad Nacional Autónoma de México, Publicaciones Preliminares, No. 24 · Zbl 0575.05031 [6] Neumann-Lara, V, The generalized dichromatic number of a digraph, (1981), Instituto de Matemáticas. Universidad Nacional Autónoma de México, Publicaciones Preliminares, No. 32 · Zbl 0575.05031
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