An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...

A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition of G. The maximum number of classes of a domatic partition of G is called the domatic number of G. The concept of a domatic number was introduced in [1]. More interesting results on domatically full graphs, domatically critical, domatically cocritical graphs and other domatic numbers can be found ...

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...

For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination...

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and m...

Let G = (V,E) be a simple undirected graph, and k be a positive integer. A k-dominating set of G is a set of vertices S ⊆ V satisfying that every vertex in V \ S is adjacent to at least k vertices in S. A k-domatic partition of G is a partition of V into k-dominating sets. The k-domatic number of G is the maximum number of k-dominating sets contained in a k-domatic partition of G. In this paper...

A dominating set S in a graph G is a tree dominating set of G if the subgraph induced by S is a tree. The tree domatic number of G is the maximum number of pairwise disjoint tree dominating sets in V (G). First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given...