Homogeneous edge-colorings of graphs (Q2831592)

Let \(G = (V, E)\) be a multigraph without loops. For any \(x \in V\), 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 \(m \geq 2\) and a coloring \(c : E \rightarrow S\) of the edges of \(G\) such that \(|S| = m\) and for any \(x \in V\), if \(|E(x)| = m q_x + r_x\) with \(0 \leq r_x < m\), 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 \({\chi}_h(G)\) is the least \(m\) for which there exists such a coloring. In this paper, the authors determined \({\chi}_h(G)\) when \(G\) is a complete multigraph, a tree, or a complete bipartite multigraph.