In this paper, we provide a new upper bound for the α-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the α-rate domination number, which combines the concepts of...

For positive integers k and d such that 4 ≤ k < d and k 6= 5, we determine the maximum number of rainbow colored copies of C4 in a k-edge-coloring of the d-dimensional hypercube Qd. Interestingly, the k-edge-colorings of Qd yielding the maximum number of rainbow copies of C4 also have the property that every copy of C4 which is not rainbow is monochromatic.

A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. A graph G is called rainbow k-connected, if there is an edge-colouring of G with k colours such that G is rainbow-connected. In this talk we will study rainbow k-connected graphs with a minimum number of edges. For an integer n ≥ 3 and 1 ≤ k ≤ n − 1 let t(n, k) denote the...

Let k be a positive integer. A vertex subset D of a graph G = (V,E) is a perfect k-dominating set of G if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γkp(G). In this paper, we generalize perfect domination to perfect k-domination, where many bounds of γkp(G) are obtained. We ...

Let G be a graph of order n ≥ 2 and n1, n2, .., nk be integers such that 1 ≤ n1 ≤ n2 ≤ .. ≤ nk and n1 + n2 + .. + nk = n. Let for i = 1, .., k: Ai ⊆ Kni where Km is the set of all pairwise non-isomorphic graphs of order m, m = 1, 2, ... In this paper we study when for a domination related parameter μ (such as domination number, independent domination number and acyclic domination number) is ful...

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number k(G), the connected k-domination number c k (G); the k-independent domination number i k (G) and the k-irredundance number irk(G). The authors prove that if an irk-set X is a k-independent set of G, then irk(G) = k(G) = k(G), and that for k ...

A set S of vertices of a graph G= (V ,E) with no isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number t(G) is the minimum cardinality of a total dominating set ofG. The total domination subdivision number sd t (G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We ...

A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The total (double, respectively) domination number of a graph G is the min...

An edge e ∈ E(G) dominates a vertex v ∈ V (G) if e is incident with v or e is incident with a vertex adjacent to v. An edge-vertex dominating set of a graph G is a set D of edges of G such that every vertex of G is edgevertex dominated by an edge of D. The edge-vertex domination number of a graph G is the minimum cardinality of an edge-vertex dominating set of G. A subset D ⊆ V (G) is a total d...