On f-colorings of nearly bipartite graphs

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 denoted by χf′(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 χf′(G)=Δf(G), then G is of f-class 1, otherwise G is of f-class 2. In this paper, we give some sufficient conditions for a nearly bipartite graph to be of f-class 1.