  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 $\tau(G)$] equals $\max\{\Delta, \omega\}$, where $\Delta$ and $\omega$
are the maximum vertex degree in $G$ and the multigraph density, respectively.
We prove that the conjecture holds iff $\tau(G)$ is a monotone function on the
set of multigraphs.
