Edge-domatic numbers of directed graphs.

*(English)*Zbl 0847.05063In [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 \(G\).

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 \(G\).

##### Keywords:

domatic number; digraphs; edge-domatic partition; edge-dominating sets; edge-domatic number; edge-domatic colourings
Full Text:
EuDML

##### 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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