Orientation-based edge-colorings and linear arboricity of multigraphs

TL;DR

This paper introduces an oriented version of $f$-colorings for multigraphs, proves a Goldberg-Seymour formula for it, and applies it to confirm the Linear Arboricity Conjecture for certain multigraphs, also establishing list-coloring equivalence.

Contribution

It develops a new oriented $f$-coloring framework, proves a Goldberg-Seymour formula for it, and applies it to advance the Linear Arboricity Conjecture for $k$-degenerate multigraphs.

Findings

The $(g,h)$-oriented chromatic index satisfies a Goldberg-Seymour formula.

The Linear Arboricity Conjecture holds for $k$-degenerate multigraphs with high maximum degree.

The $(g,h)$-oriented chromatic index equals its list coloring version.

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 $k$-degenerate 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.