Some graphs of class 1 for ff-colorings

An ff-coloring of a multigraph GG is an edge-coloring of GG 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 of GG and denoted by χf′(G). Any graph GG has the ff-chromatic index equal to Δf(G)Δf(G) or Δf(G)+1Δf(G)+1, where Δf(G)=maxv∈V(G){⌈d(v)/f(v)⌉}Δf(G)=maxv∈V(G){⌈d(v)/f(v)⌉}. If χf′(G)=Δf(G), then GG is of ff-class 1, and otherwise GG is of ff-class 2. In this work, on the basis of the ff-core of GG (i.e., the subgraph of GG induced by the vertices of V0∗(G)={v:Δf(G)=d(v)/f(v),v∈V(G)}), we give some sufficient conditions for a graph GG to be of ff-class 1.