No abstract

If the availability of colors to color a graph [Formula: see text] is less than that of the chromatic number [Formula: see text] of the graph, then coloring the graph with available colors, say [Formula: see text] colors, where [Formula: see text], will cause the end vertices of at least one edge to be colored with same color. Such an edge whose end vertices receive a same color is called as a bad edge. A coloring that restricts few color classes to have adjacency between the elements in it so as to minimize the number of bad edges obtained from it in a graph [Formula: see text] is called as a near proper coloring and a near proper coloring that uses [Formula: see text] colors where [Formula: see text] to color a graph [Formula: see text] by permitting only one color class to have adjacency among the elements in it and thereby minimize the number of bad edges resulting from the permitted color class is called as a [Formula: see text]-coloring of the graph [Formula: see text]. In this paper, we determine the number of bad edges of powers of helm graphs [Formula: see text] and powers of closed helm graphs [Formula: see text].

The chromatic number, [Formula: see text] of a graph [Formula: see text] is the minimum number of colors used in a proper coloring of [Formula: see text]. In an improper coloring, an edge [Formula: see text] is bad if the colors assigned to the end vertices of the edge is the same. Now, if the available colors are less than that of the chromatic number of graph [Formula: see text], then coloring the graph with the available colors leads to bad edges in [Formula: see text]. In this paper, we use the concept of [Formula: see text]-coloring and determine the number of bad edges in generalized Petersen graph ([Formula: see text]). The number of bad edges which result from a [Formula: see text]-coloring of [Formula: see text] is denoted by [Formula: see text].

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].