f-Class two graphs whose f-cores have maximum degree two

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex . . V (G) at most f(.) times. The f-core of G is the subgraph of G induced by the vertices . of degree d(v)=f(v)maxv.V(G){.d(v)/f(v).}. In this paper, we find some necessary conditions for a simple graph, whose f-core has maximum degree two, to be of class 2 for f-colorings.