Nearly optimal packings of equally sized rainbow forests

Abstract

A forest in an edge-colored graph is rainbow if its edges have pairwise distinct colors. We prove that for every $\varepsilon>0$ and all sufficiently large integers $m$, every properly edge-colored simple graph with $km$ edges, where $1\leq k\leq 2m$ and every color class has size at most $m$, contains at least $(1-\varepsilon)m$ pairwise edge-disjoint rainbow forests, each with exactly $k$ edges. The range $k\leq 2m$ is best possible: for every $k>2m$ there are such graphs containing no $k$-edge forest. Thus the conjecture of Montgomery, Pokrovskiy, and Sudakov fails beyond this range, while our theorem establishes its predicted conclusion throughout the largest possible range of $k$. The number of forests is asymptotically optimal. The proof uses an orientation dichotomy, hypergraph matching, matroid intersection, and martingale concentration.