Some sufficient conditions for a graph to be of image 1 Original Research Article

An ff-coloring of a graph GG is a coloring of edges of E(G)E(G) such that each color appears at each vertex v∈V(G)v∈V(G) at most f(v)f(v) times. The minimum number of colors needed to ff-color GG is called the ff-chromatic index View the MathML sourceχf′(G) of GG. Any graph GG has ff-chromatic index equal to Δf(G)Δf(G) or Δf(G)+1Δf(G)+1, where View the MathML sourceΔf(G)=maxv∈V{⌈d(v)f(v)⌉}. If View the MathML sourceχf′(G)=Δf(G), then GG is of CfCf 1; otherwise GG is of CfCf 2. Some sufficient conditions for a graph to be of CfCf 1 are given.