No abstract

Let [Formula: see text] be the minimum number of distinct resources or equipment such as channels, transmitters, antennas and surveillance equipment required for a system’s stability. These resources are placed on a system. The system is stable only if the resources of the same type are placed far away from each other or, in other words, they are not adjacent to each other. Let these distinct resources represent different colors assigned on the vertices of a graph [Formula: see text]. Suppose the available resources, denoted by [Formula: see text], are less than [Formula: see text]. In that case, placing [Formula: see text] resources on the vertices of [Formula: see text] will make at least one equipment of the same type adjacent to each other, which thereby make the system unstable. In [Formula: see text]-coloring, the adjacency between the resources of a single resource type is tolerated. The remaining resources are placed on the vertices so that no two resources of the same type are adjacent to each other. In this paper, we discuss some general results on the [Formula: see text]-coloring and the number of bad edges obtained from the same for a graph [Formula: see text]. Also, we determine the minimum number of bad edges obtained from [Formula: see text]-coloring of few derived graph of graphs. The number of bad edges which result from a [Formula: see text]-coloring of [Formula: see text] is denoted by [Formula: see text].

A set of vertices in a graph is a dominating set if every vertex not in the set is adjacent to at least one vertex in the set. A dominating structure is a subgraph induced by the dominating set. Connected domination is a type of domination where the dominating structure is connected. Clique domination is a type of domination in graphs where the dominating structure is a complete subgraph. The clique domination number of a graph G denoted by γk(G) is the minimum cardinality among all the clique dominating sets of G. We present few properties of graphs admitting dominating cliques along with bounds on clique domination number in terms of order and size of the graph. A necessary and sufficient condition for the existence of dominating clique in strong product of graphs is presented. A forbidden subgraph condition necessary to imply the existence of a connected dominating set of size four also is found.