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The Goldberg-Seymour Conjecture for $f$-colorings states that the$f$-chromatic index of a loopless multigraph is essentially determined byeither a maximum degree or a maximum density parameter. We introduce anoriented version of $f$-colorings, where now each color class of theedge-coloring is required to be orientable in such a way that every vertex $v$has indegree and outdegree at most some specified values $g(v)$ and $h(v)$. Weprove that the associated $(g,h)$-oriented chromatic index satisfies aGoldberg-Seymour formula. We then present simple applications of this result tovariations of $f$-colorings. In particular, we show that the Linear ArboricityConjecture holds for $k$-degenerate loopless multigraphs when the maximumdegree is at least $4k-2$, improving a bound recently announced by Chen, Hao,and Yu for simple graphs. Finally, we demonstrate that the $(g,h)$-orientedchromatic index is always equal to its list coloring analogue.

Orientation-based edge-colorings and linear arboricity of multigraphs