On k-ply domatic numbers of graphs.

*(English)*Zbl 0602.05039A k-ply dominating set in an undirected graph G is a subset D of the vertex set V(G) of G with the property that for each vertex \(x\in V(G)-D\) there exist k pairwise distinct vertices \(y_ 1,...,y_ k\) which are all adjacent to x. The maximum number of classes of a partition of V(G) into k-ply dominating sets is called the k-ply domatic number of G and denoted by \(d^ k(G)\). This is a generalization of the domatic number of a graph which was introduced by E. J. Cockayne and S. T. Hedetniemi. The properties of the k-ply domatic number of a graph are described. Some inequalities are presented and the interrelations among k-ply domatic numbers for different numbers k are treated. The k-ply domatic numbers of circuits and of complete bipartite graphs are stated. At the end edge- critical graphs with respect to the k-ply domatic number are described.

##### MSC:

05C35 | Extremal problems in graph theory |

##### Keywords:

vertex partition; k-ply dominating set; k-ply domatic number; circuits; complete bipartite graphs; edge-critical graphs
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##### References:

[1] | COCKAYNE E. J., HEDETNIEMI S. T.: Towards a theory of domination in graphs. Network s7, 1977, 247-261. · Zbl 0384.05051 |

[2] | ZELINKA B.: Domatically critical graphs. Czech. Math. J. 30, 1980, 486-489. · Zbl 0426.05046 |

[3] | ZELINKA B.: On k-domatic numbers of graphs. Czech. Math. J. 33, 1983, 309-313. · Zbl 0537.05050 |

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