In this paper we investigate the dominating- -color number،  of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and H. This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].  

Dominating -color number of a graph is defined as the maximum number of color classes which are dominating sets of and is denoted by d, where the maximum is taken over all -coloring of . In this paper, we discussed the dominating -color number of Generalized Petersen Graphs. We have also discussed the condition under which chromatic number equals dominating -color number of Generalized Pet...

Graph coloring is used to characterize some properties of graphs. A b-coloring of a graph G (using colors 1,2,...,k) is a coloring of the vertices of G such that (i) two neighbors have different colors (proper coloring) and (ii) for each color class there exists a dominating vertex which is adjacent to all other k-1 color classes. In this paper, we build on b-coloring of a graph to propose a ne...

Let G = (V, E) be a graph. We define strong dominating color number of a graph G as the maximum number of color classes which are strong dominating sets of G, and is denoted by sd(G) , where the maximum is taken over all  -coloring of G. This paper determines the exact values of strong dominating color number of sum and Cartesian product of graphs.

In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different col...

The Square Coloring of a graph G refers to coloring vertices such that any two distinct which are at distance most receive different colors. In this paper, we initiate the study related problem called subset square graphs. Broadly, studies dominating set using q Here aim is optimize number colors used. This also generalizes well-studied Efficient Dominating Set problem. We show $$q$$ -Subset NP...

The domatic partition problem seeks to maximize the partitioning of the nodes of the network into disjoint dominating sets. These sets represent a series of virtual backbones for wireless sensor networks to be activated successively, resulting in more balanced energy consumption and increased network robustness. In this study, we address the domatic partition problem in random geometric graphs ...

Clustering algorithms consist in automatically discovering structure of large data set or providing coherent groups of objects independent of any user-defined classes. They are used in many domains: astronomy, information retrieval, image segmentation, biological applications and so on. Several clustering techniques have been proposed [1], there are either hierarchical techniques or partitionin...

A dominating coloring by k colors is a proper k coloring where every color i has a representative vertex xi adjacent to at least one vertex in each of the other classes. The b-chromatic number, b(G), of a graph G is the largest integer k such that G admits a dominating coloring by k colors. A graph G = (V,E) is said b-monotonous if b(H1) ≥ b(H2) for every induced subgraph H1 of G and every subg...