Packing large balanced trees into bipartite graphs

TL;DR

This paper proves that large families of balanced trees can be embedded into complete bipartite graphs, extending packing results and introducing an approximate bipartite version of a key combinatorial theorem.

Contribution

It establishes a new packing theorem for large families of balanced trees into bipartite graphs and develops an approximate bipartite version of the Komlós-Sárközy-Szemerédi Theorem.

Findings

Families of up to n^{1/2+γ} trees can be packed into K_{n,n}

Introduces an approximate bipartite version of a fundamental theorem

Provides tools of independent interest for bipartite graph packing

Abstract

We prove that for every ${\gamma > 0}$ there exists $n_0 \in \mathbb{N}$ such that for every ${n \geq n_0}$ any family of up to $\lfloor{n^{\frac12+\gamma}}\rfloor$ trees having at most $(1-\gamma)n$ vertices in each bipartition class can be packed into $K_{n,n}$. As a tool for our proof, we show an approximate bipartite version of the Koml\'os-S\'ark\"ozy-Szemer\'edi Theorem, which we believe to be of independent interest.