Homogeneous EDGE-colorings of graphs

Abstract

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 ∈f |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 - rx color classes of cardinality q_x. The homogeneous chromatic index χ(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.