On an $f$-coloring generalization of linear arboricity of multigraphs

Abstract

Given a multigraph $G$ and function $f : V(G) \rightarrow \mathbb{Z}_{\ge 2}$ on its vertices, a degree-$f$ subgraph of $G$ is a spanning subgraph in which every vertex $v$ has degree at most $f(v)$. The degree-$f$ arboricity $a_f(G)$ of $G$ is the minimum number of colors required to edge-color $G$ into degree-$f$ forests. At least for constant $f$, Truszczy\'nski conjectured that $a_f(G) \le \max \{\Delta_f(G) + 1, a(G)\}$ for every multigraph $G$, where $\Delta_f(G) = \max_{v \in V(G)} \lceil d(v)/f(v) \rceil$ and $a(G)$ is the usual arboricity of $G$. This is a strong generalization of the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary. In this paper, we disprove Truszczy\'nski's conjecture in a strong sense for general multigraphs. On the other hand, extending known results for linear arboricity, we prove that the conjecture holds for simple graphs with sufficiently large girth, and that it holds for all simple graphs asymptotically. More strongly, we prove these partial results in the setting of directed graphs, where the color classes are required to be analogously defined degree-$f$ branchings.