Decomposing a graph into forests
Résumé
In this paper, we propose a conjecture which says that every graph $G$ with $\gamma_f(G) \le k+ \frac{d}{k+d+1}$ decomposes into $k+1$ forests, and one of the forests has maximum degree at most $d$. We prove two special cases of this conjecture: If $G$ is a graph with fractional arboricity at most $\frac 43$, then $G$ decomposes into a forest and a matching. If $G$ is a graph with fractional arboricity at most $\frac 32$, then $G$ decomposes into a forest and a linear forest. In particular, every planar graph of girth at least $8$ decomposes into a forest and a matching, and every planar graph of girth at least $6$ decomposes into a forest and a linear forest. This improves earlier results concerning decomposition of planar graphs. We give examples to show that there are planar graphs of girth $7$ which does not decompose into a forest and a matching, and there are planar graphs of girth $5$ which does not decompose into a forest and a linear forest. We also show that the bound in the conjecture above is sharp, i.e., for any $\epsilon > 0$, there is a graph $G$ with $\gamma_f(G) < k+ \frac{d}{k+d+1} +\epsilon$, and yet $G$ cannot be decomposed into $k+1$ forests, with one of them having maximum degree at most $d$.