Orientation-based edge-colorings and linear arboricity of multigraphs

Abstract

The Goldberg-Seymour Conjecture for f -colorings states that the f -chromatic index of a loopless multigraph is essentially determined by either a maximum degree or a maximum density parameter. We introduce an oriented version of f -colorings, where now each color class of the edge-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). We prove that the associated (g, h)-oriented chromatic index satisfies a Goldberg-Seymour formula. We then present simple applications of this result to variations of f -colorings. In particular, we show that the Linear Arboricity Conjecture holds for kdegenerate loopless multigraphs when the maximum degree 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)-oriented chromatic index is always equal to its list coloring analogue.