On δ(k)-coloring of powers of helm and closed helm graphs

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].