An extension of bipartite multigraphs

Usual edge colorings have been generalized in various ways; we will consider here essentially good edge colorings as well as equitable edge colorings. It is known that bipartite multigraphs present the property of having an equitable k-coloring for each k ? 2. This implies that they also have a good k-coloring for each k ? 2. In this paper, we characterize a class of multigraphs which may be considered as a generalization of bipartite multigraphs, in the sense that for each k ? 2 they have a good k-coloring. A more restrictive class is derived where all multigraphs have an equitable k-coloring for each k ? 2.