Hi everyone,

I have a problem I am working on that can be reduced to the following special case of edge coloring.

Let $G = (V,E)$ be an arbitrary graph. Furthermore, let each edge be assigned a positive integer so that we have function $w: E \rightarrow  N$. Let $C$ be a set of colors, represented by an interval of integers. Can we assign a coloring to the edges of $G$ so that for each each $e$, if $w(e) = a$ then $e$ receives $a$ colors, the colors assigned to $e$ form an interval of the color set, and no edge shares a color with another incident edge. (Alteratively, is there an approximation factor similar to the one given Vizing's theorem for the standard edge-coloring problem).

I have done a bunch of literature searches and have already discovered that this problem is not the same as: interval edge coloring (close but not that close) and weighted edge coloring (closer but generally only applicable to bipartite graphs).

> Has anyone seen this problem before? Are there any results? Would you
> recommend any references or perhaps additional directions to search.

Many thanks,
Scott