Orientation‐based edge‐colorings and linear arboricity of multigraphs
Ronen Wdowinski · Journal of Graph Theory · 2022
Abstract The Goldberg–Seymour Conjecture for ‐colorings states that the ‐chromatic index of a loopless multigraph is essentially determined by either a weighted maximum degree or a weighted maximum density parameter. We introduce an oriented version of ‐colorings, where now each color class of the edge‐coloring is required to be orientable in such a way that every vertex has indegree and outdegree at most some specified values and . We prove that the associated ‐oriented chromatic index satisfies a Goldberg–Seymour formula. We then present simple applications of this result to variations of ‐colorings. In particular, we show that the Linear Arboricity Conjecture holds for ‐degenerate loopless multigraphs when the maximum degree is at least , improving a recent bound by Chen, Hao, and Yu for simple graphs. Finally, we demonstrate that the ‐oriented chromatic index is always equal to its list coloring analogue.