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Dichromatic number, circulant tournaments and Zykov sums of digraphs. (English) Zbl 0984.05043
Author’s abstract: The dichromatic number $$\text{dc}(D)$$ of a digraph $$D$$ is the smallest number of colors needed to color the vertices of $$D$$ so that no monochromatic directed cycle is created. In this paper the problem of computing the dichromatic number of a Zykov sum of digraphs over a digraph $$D$$ is reduced to that of computing a multicovering number of a hypergraph $$H_1(D)$$ associated to $$D$$ in a natural way. This result allows us to construct an infinite family of pairwise non-isomorphic vertex-critical $$k$$-dichromatic circulant tournaments for every $$k\geq 3$$, $$k\neq 7$$.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C15 Coloring of graphs and hypergraphs 05C65 Hypergraphs
##### Keywords:
dichromatic number; digraph; Zykov sum; circulant tournaments
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