On degree-colorings of multigraphs

A notion of degree-coloring is introduced; it captures some, but not all properties of standard edge-coloring. We conjecture that the smallest number of colors needed for degree-coloring of a multigraph $G$ [the degree-coloring index $τ(G)$] equals $\max\{Δ, ω\}$, where $Δ$ and $ω$ are the maximum vertex degree in $G$ and the multigraph density, respectively. We prove that the conjecture holds iff $τ(G)$ is a monotone function on the set of multigraphs.

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