On ?(k)-coloring of powers of helm and closed helm graphs
bad edges; Improper coloring; near proper coloring; k)-coloring
If the availability of colors to color a graph G is less than that of the chromatic number ?(G) of the graph, then coloring the graph with available colors, say k colors, where 1 ? k ? ?(G)-1, 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 G is called as a near proper coloring and a near proper coloring that uses k colors where 1 ? k ? ?(G)-1 to color a graph G 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 ?(k)-coloring of the graph G. In this paper, we determine the number of bad edges of powers of helm graphs H1,nr and powers of closed helm graphs CH1,nr. 2022 World Scientific Publishing Company.
Discrete Mathematics, Algorithms and Applications, Vol-14, No. 2
Ellumkalayil M.T., Department of Mathematics, CHRIST (Deemed to be University), Karnataka, Bangalore, 560029, India; Naduvath S., Department of Mathematics, CHRIST (Deemed to be University), Karnataka, Bangalore, 560029, India