## Total domatic number and degrees of vertices of a graph.(English)Zbl 0688.05066

The total domatic number of a graph G, denoted by $$d_ t(G)$$ is defined. A result concerning a lower bound for $$d_ t(G)$$, in terms of $$\delta$$ (G) and the number of vertices of G, is proved.
If G is a finite graph with n ($$\geq 9)$$ vertices and $$\delta (G)=n-3$$, it is shown that $$d_ t(G)=[n/2].$$ It is further shown that $$d_ t(G)<[n/2]$$ if $$3\leq n\leq 8$$.
Reviewer: I.H.Nagaraja Rao

### MSC:

 05C99 Graph theory

### Keywords:

dominating sets; total domatic number
Full Text:

### References:

 [1] COCKAYNE E. J., HEDETNIEMI S. T.: Towards a theory of domination in graphs. Networks 7, 1977,247-261. · Zbl 0384.05051 [2] COCKAYNE E. J., DAWES R. M., HEDETNIEMI S. T.: Total domination in graphs. Networks 10, 1980, 211-219. · Zbl 0447.05039 [3] ZELINKA B.: Domatic number and degrees of vertices of a graph. Math. Slovaca 33, 1983, 145-147. · Zbl 0516.05032 [4] ZELINKA B.: Graphs with the domatic number $$\aleph_0$$. Czech. Math. J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.