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The 3 and 4-dichromatic tournaments of minimum order. (English) Zbl 0829.05028
Author’s abstract: The dichromatic number $$\text{dc} (D)$$ of a digraph $$D$$ is the minimum number of acyclic sets in which $$V(D)$$ can be partitioned. If $$\text{dc}(D) = r$$, $$D$$ is called $$r$$-dichromatic. It is proved that the minimum order of a 3-dichromatic (resp. 4-dichromatic) tournament is 7 (resp. 11). It is also proved that there are exactly four nonisomorphic 3-dichromatic tournaments of order 7 and a unique 4- dichromatic tournament of order 11. All these tournaments are characterized.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C20 Directed graphs (digraphs), tournaments 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
##### Keywords:
isomorphism; dichromatic number; digraph; minimum order; tournament
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##### References:
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