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Edge-domatic numbers of directed graphs. (English) Zbl 0847.05063
In [Networks 7, 247-261 (1977; Zbl 0384.05051)] E. J. Cockayne and S. T. Hedetniemi introduced the domatic number of an undirected graph $$G$$ as the maximum number of classes of a partition of the vertex set of $$G$$ into dominating sets. Many variants of this number have been later studied, among them the edge-domatic number of an undirected graph [B. Zelinka, Czech. Math. J. 33(108), 107-110 (1983; Zbl 0537.05049)]. Here we will study an analogous concept for directed graphs. The adjacency of edges in a directed graph will be introduced analogously to our paper [Edge-chromatic numbers of directed graphs (to appear)]. We consider finite directed graphs (shortly digraphs) without loops in which two vertices may be joined by two edges only if these edges are oppositely directed. Two edges of a digraph $$G$$ will be called adjacent, if the terminal vertex of one of them is the initial vertex of the other. A subset $$D$$ of the edge set $$E(G)$$ of $$G$$ is called edge-dominating, if for each edge $$e \in E(G) - D$$ there exists an edge $$f \in D$$ adjacent to $$e$$. A partition of $$E(G)$$ is called an edge-domatic partition of $$G$$, if all of its classes are edge-dominating sets in $$G$$. The maximum number of classes of an edge-domatic partition of $$G$$ is called the edge-domatic number of $$G$$ and denoted by $$\text{ed} (G)$$.
Sometimes it is more convenient to speak about edge-domatic colourings instead of edge-domatic partitions. A colouring of edges of a digraph $$G$$ is called edge-domatic, if each edge is adjacent in $$G$$ to edges of all colours different from its own. (Two adjacent edges may be coloured by the same colour). Then the edge-domatic number of $$G$$ is equal to the maximum number of colours of an edge-domatic colouring of $$G$$. Equivalence of this definition with the previous one is evident. The edge domatic number of a directed graph $$G$$ is evidently equal to the domatic number of the graph $$L(G)$$ whose vertex set is is the edge set of $$G$$ and in which two vertices are adjacent if and only if they are adjacent as edges in $$G$$.

##### MSC:
 05C35 Extremal problems in graph theory 05C20 Directed graphs (digraphs), tournaments
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##### References:
 [1] E. J. Cockayne, S. T. Hedetniemi: Towards a theory of domination of graphs. Networks 7 (1977), 247-261. · Zbl 0384.05051 · doi:10.1002/net.3230070305 [2] B. Zelinka: Edge-domatic number of a graph. Czech. Math. J. 33 (1983), 107-110. · Zbl 0537.05049 · eudml:13366 [3] B. Zelinka: Edge-chromatic numbers of directed graphs. Math. Slovaca. · Zbl 0847.05063 · eudml:31478
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