Packing large trees of consecutive orders

TL;DR

This paper proves Bollobás's conjecture for packing large trees with specific leaf or path structures into complete graphs, confirming the conjecture for up to five trees and advancing understanding of tree packings.

Contribution

The paper confirms Bollobás's conjecture for trees with certain leaf or path configurations, extending known cases and verifying the conjecture for up to five trees.

Findings

Confirmed the conjecture for trees with specific leaf or path structures.

Validated the conjecture for any set of up to five trees.

Provided new techniques for tree packing problems.

Abstract

A conjecture by Bollob\'as from 1995 (which is a weakenning of the famous Tree Packing Conjecture by Gy\'arf\'as from 1976) states that any set of $k$ trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $n-i$ vertices, pack into $K_n$, provided $n$ is sufficiently large. We confirm Bollob\'as conjecture for trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $k-1-i$ leaves or a pending path of order $k-1-i$. As a consequence we obtain that the conjecture is true for $k\leq 5$.