We prove that for any positive integer k, the edges of any graph whose fractional arboricity is at most k+1/(3k+2) can be decomposed into k forests and a matching.
@article{arxiv.1007.0316,
title = {Covering a graph by forests and a matching},
author = {Tomas Kaiser and Mickael Montassier and Andre Raspaud},
journal= {arXiv preprint arXiv:1007.0316},
year = {2010}
}