Induced and Weak Induced Arboricities

TL;DR

This paper introduces the concept of induced arboricity, explores its bounds across various graph classes, and relates it to shallow minors and chromatic numbers, providing new insights into graph covering properties.

Contribution

It defines induced arboricity and establishes bounds for it across different graph classes, linking it to shallow minors and chromatic numbers, and introduces weak and star induced arboricities.

Findings

Bounded induced arboricity for planar graphs between 8 and 10.

Induced arboricity is finite iff the class's shallow minors have finite chromatic number.

Introduces bounds for weak and star induced arboricities.

Abstract

We define the induced arboricity of a graph $G$, denoted by ${\rm ia}(G)$, as the smallest $k$ such that the edges of $G$ can be covered with $k$ induced forests in $G$. This notion generalizes the classical notions of the arboricity and strong chromatic index. For a class $\mathcal{F}$ of graphs and a graph parameter $p$, let $p(\mathcal{F}) = \sup\{p(G) \mid G\in \mathcal{F}\}$. We show that ${\rm ia}(\mathcal{F})$ is bounded from above by an absolute constant depending only on $\mathcal{F}$, that is ${\rm ia}(\mathcal{F})\neq\infty$ if and only if $\chi(\mathcal{F} \nabla \frac{1}{2}) \neq\infty$, where $\mathcal{F} \nabla \frac{1}{2}$ is the class of $\frac{1}{2}$-shallow minors of graphs from $\mathcal{F}$ and $\chi$ is the chromatic number. Further, we give bounds on ${\rm ia}(\mathcal{F})$ when $\mathcal{F}$ is the class of planar graphs, the class of $d$-degenerate graphs, or the class of graphs having tree-width at most $d$. Specifically, we show that if $\mathcal{F}$ is the class of planar graphs, then $8 \leq {\rm ia}(\mathcal{F}) \leq 10$. In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.