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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...
A tournament is said to be tight whenever every 3-colouring of its vertices using the 3 colours, leaves at least one cyclic triangle all whose vertices have different colours. In this paper, we extend the class of known tight circulant tournaments.
The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes.
For any two integers s and n with 1 ≤ s ≤ n, let be the oriented graph such that is the set of integers mod 2n+1 and In this paper we prove that for n ≥ 7. The bound is tight since equality holds when s ∈ n,[(2n+1)/3].
If G is a minimally 3-connected graph and C is a double cover of the set of edges of G by irreducible walks, then |E(G)| ≥ 2| C| - 2.
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