On Edge Cover Colorings Of Graphs

Let G(V,E) be a simple graph with edge set E(G) and vertex set V (G).A mapping C of G from E(G) to a color set{1,2,…,K} is called a h-edge-coloring of G. Let (i) denote the number of edges of G incident withvertex:receiving color i by the coloringC .If for each ,and {1, 2,…,k}, C is called a h-edge cover-coloring of G. Let (G)be the maximum positive integer h for which an h-edge cover-coloring ofG exists. Then (G) is called the edge cover chromatic index of G. Itis known that G is a graph of CIclass, otherwise G is a graph of CII class. In this paper,we consider theclassification of Halin graphs based on the edge cover chromatic index. Weobtain a necessary and sufficient condition for some Hafin graphs to be ofclass CI; We also obtain a property of edge cover critical graphs.