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On k-ply domatic numbers of graphs. (English) Zbl 0602.05039
A 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
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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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