f-Colorings of some graphs of f-class 1

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v ∈ V(G) at most f(v) times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by x′f(G). Any simple graph G has the f-chromatic index equal to Δf(G) or Δf(G) + 1, where Δf(G) = maxv∈V(G){⌈d(v)/f(v)⌊}. If x′f(G) = Δf(G), then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with Δf(G) colors are given.