NK-Labelling of Some Graphs

Abstract

Graph , P is a number verticese of    and q number of edges of Graph labelling and coloring form one of the most active branches of modern graph theory, with wide-ranging applications in scheduling, network design, coding theory, and frequency assignment. In this study we report the variant of edge coloingthat we call NK-labelling (NK-edge coloring), in which the  proper  edge coloring of the graph is a  required from additionally satisfy a neighborhood condition: every vertex must see each color used at most once among its incident edges, so that the coloring is simultaneously proper and “neighborhood-consistent.” We formalize the associated parameter, the NK-chromatic index χ′_{NK}(G).  In this paper we  investigated some graphs  : K_n with n odd and to other graph families such as cycles, complete bipartite graphs, and Cartesian products are studied also.