Homogeneous edge-colorings of graphs

Let G = (V, E) be a multigraph without loops and for any x ∈ V let E(x) be the set of edges of G incident to x. A homogeneous edge-coloring of G is an assignment of an integer m ≥ 2 and a coloring c∶ E → S of the edges of G such that ∣S∣ = m and for any x ∈ V , if ∣E(x)∣ = mq x + r x with 0 ≤ r x < m, there exists a partition of E(x) in r x color classes of cardinality q x + 1 and other m − r x color classes of cardinality q x . The homogeneous chromatic index x ̃(G) is the least m for which there exists such a coloring. We determine χ ̃ (G) in the case that G is a complete multigraph, a tree or a complete bipartite multigraph.