On δ(k)-Coloring of Some Cycle-Related Graphs

Abstract

A graph coloring is proper when the colors assigned to a pair of adjacent vertices in it are different and it is improper when at least one of the adjacent pair of vertices receives the same color. When the minimum number of colors required in a proper coloring of a graph is not available, coloring the graph with the available colors, say [Formula: see text] colors, will lead at least an edge to have its end vertices colored with a same color. Such an edge is called a bad edge. In a proper coloring of a graph [Formula: see text], every color class is an independent set. However, in an improper coloring there can be few color classes that are non-independent. In this paper, we use the concept of [Formula: see text]-coloring, which permits only one color class to be non-independent and determine the minimum number of bad edges, which is denoted by [Formula: see text], obtained from the same for some cycle-related graphs.