On Decomposing Graphs Into Forests and Pseudoforests

Abstract

We prove that for $k\in \mathbb{N}$ and $d\le 2k+2$, if a graph has maximum average degree at most $2k+\frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of the pseudoforests has all connected components having at most $d$ edges.

12 pages, 2 figures. This was submitted to the Canadam prize competition. We refer the reader arXiv:1905.02600 for a better result and exposition