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Packing large balanced trees into bipartite graphs

Cristina G. Fernandes, Tássio Naia, Giovanne Santos, Maya Stein

Abstract

We prove that for every ${γ> 0}$ there exists $n_0 \in \mathbb{N}$ such that for every ${n \geq n_0}$ any family of up to $\lfloor{n^{\frac12+γ}}\rfloor$ trees having at most $(1-γ)n$ 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 Komlós-Sárközy-Szemerédi Theorem, which we believe to be of independent interest.

Packing large balanced trees into bipartite graphs

Abstract

We prove that for every there exists such that for every any family of up to trees having at most vertices in each bipartition class can be packed into . As a tool for our proof, we show an approximate bipartite version of the Komlós-Sárközy-Szemerédi Theorem, which we believe to be of independent interest.

Paper Structure

This paper contains 13 sections, 10 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Ringel-type conjectures
  3. Gyárfás' Tree Packing Conjecture
  4. Ringel-type tree-packing for balanced trees
  5. A bipartite Komlós--Sárközy--Szemerédi Theorem
  6. Preliminaries
  7. Regularity
  8. Tree partitioning and assigning results
  9. Embedding trees in dense balanced graphs: proof of Theorem \ref{['thm:emb_rooted_tree']}
  10. Packing large balanced trees: Proof of Theorem \ref{['thm:packing_trees']}
  11. Final Remarks
  12. Open questions for tree-packing into bipartite graphs
  13. An exact bipartite KSS theorem?

Key Result

Theorem 7

For every $\gamma > 0$, there exists $n_0$ such that for every ${n \geq n_0}$ any family of at most $n^{\frac{1}{2}-\gamma}$ rooted trees, each with at most $(1-\gamma)n$ vertices in either partition class, packs into $K_{n,n}$, with all tree roots embedded in the same part of $K_{n,n}$.

Theorems & Definitions (22)

  • Conjecture 1: Ringel R63-theory_of_graphs
  • Conjecture 2: Böttcher, Hladký, Piguet and Taraz BHPT16-approximate_tpc
  • Conjecture 3: Graham and Häggkvist H89-decompositions_bipartite
  • Conjecture 4: Gyárfás' Tree Packing Conjecture GL76-trees_into_kn
  • Conjecture 5: Hobbs, Bourgeois, and Kasiraj HBK87-trees_complete_graphs
  • Conjecture 6: Hollingsworth H13-balanced_trees
  • Theorem 7
  • Theorem 8
  • Corollary 9
  • Theorem 10: Komlós, Sárközy and Szemerédi KSS01-spanning_trees
  • ...and 12 more