Decomposing a graph into forests

Abstract

The fractional arboricity γ f ( G ) of a graph G is the maximum of the ratio | E ( G [ X ] ) | / ( | X | − 1 ) over all the induced subgraphs G [ X ] of G. In this paper, we propose a conjecture which says that every graph G with γ f ( G ) ⩽ k + 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 4 3 , then G decomposes into a forest and a matching. If G is a graph with fractional arboricity at most 3 2 , 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, and the girth condition is sharp, as there are planar graphs of girth 7 which do not decompose into a forest and a matching, and there are planar graphs of girth 5 which do not decompose into a forest and a linear forest. The bound in the conjecture above is sharp: We shall show that for any ϵ > 0 , there is a graph G with γ f ( G ) < k + d k + d + 1 + ϵ , and yet G cannot be decomposed into k forests plus a graph of maximum degree at most d. On the other hand, we show that for any positive integer k and real number 0 ⩽ ϵ < 1 , every graph G with γ f ( G ) ⩽ k + ϵ decomposes into k forests plus a graph of maximum degree at most ⌈ ( k + 1 ) ( k − 1 + 2 ϵ ) ( 1 − ϵ ) ⌉ .