On the Total {k}-Domination and Total {k}-Domatic Number of Graphs

作者：Aram H^*; Sheikholeslami S M; Volkmann L

来源：Bulletin of the Malaysian Mathematical Sciences Society, 2013, 36(1): 39-47.

• Total {k}-dominating function
• total {k}-domination number
• total {k}-domatic number

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function from the vertex set V (G) to the set {0,1,2, . . . ,0} such that for any vertex v is an element of V (G), the condition Sigma(u is an element of N(v)) f (u) >= k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value omega(f) = Sigma(v is an element of v) f(v). The total {k}-domination number, denoted by gamma({k})(t) (G), is the minimum weight of a total {k}-dominating function on G. A set {f(1),f(2),...,f(d)} of total {k}-dominating functions on G with the property that Sigma(d)(i=1) fi(v) <= k for each v is an element of V (G), is called a total {k} -dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d(t)({k}) (G). Note that d(t)({1}) (G) is the classic total domatic number d(t) (G). In this paper, we present bounds for the total {k}-domination number and total {k}-domatic number. In addition, we determine the total {k}-domatic number of cylinders and we give a Nordhaus-Gaddum type result.

• 出版日期2013