DMGT

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ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 20(2) (2000) 197-207
DOI: 10.7151/dmgt.1119

DICHROMATIC NUMBER, CIRCULANT TOURNAMENTS AND ZYKOV SUMS OF DIGRAPHS

Víctor Neumann-Lara

Instituto de Matemáticas, UNAM
Circuito Exterior, C.U.
México 04510 D.F., MÉXICO
e-mail: neumann@matem.unam.mx

Abstract

The dichromatic number dc(D) of a digraph D is the smallest number of colours needed to colour 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 an hypergraph H1(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 ≥ 3, k ≠ 7.

Keywords: digraphs, dichromatic number, vertex-critical, Zykov sums, tournaments, circulant, covering numbers in hypergraphs.

2000 Mathematics Subject Classification: 05C20, 05C15, 05C65.

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Received 10 November 1999
Revised 30 October 2000