Packing the largest trees in the tree packing conjecture

Abstract

The famous tree packing conjecture of Gyárfás from 1976 says that any sequence of trees $T_1,\ldots,T_n$ such that $|T_i|=i$ for each $i\in [n]$ packs into the complete $n$-vertex graph $K_n$. Packing even just the largest trees in such a sequence has proven difficult, with Bollobás drawing attention to this in 1995 by conjecturing that, for each $k$, if $n$ is sufficiently large then the largest $k$ trees in any such sequence can be packed into $K_n$. This has only been shown for $k\leq 5$, by Żak, despite many partial results and much related work on the full tree packing conjecture. We prove Bollobás's conjecture, by showing that, moreover, a linear number of the largest trees can be packed in the tree packing conjecture.

Figures (5)

• Figure 1: Grids of vertices in $W_i$, $i\in [7]$, corresponding to the sequences $P_1,S_1,S_2,P_2,S_3,S_4,P_3,P_4,S_5,S_6,P_5,S_7,P_6,P_7,S_8,S_9,S_{10},S_{11}$ (left) and $S_1,P_1,P_2,P_3,S_2,S_3,S_4,P_4,S_5,S_6,P_5,S_7,P_6,S_8,S_9,P_7,S_{10},S_{11}$ (right).
• Figure 2: The order of the vertices (mostly) covered by the embedding. Firstly, part of $P_i$ is embedded in $W_i\setminus W_{i+1}$, for each even $i$ (\ref{['stepA']}). Then, $P_1$ is embedded covering $W_1$ (\ref{['stepB']}). Then, for each even $i$, $P_i$ and $P_{i+1}$ are embedded together covering most of $W_i\setminus W_{i+1}$ (\ref{['stepC']}).
• Figure 3: Steps \ref{['stepA']} and \ref{['stepB']} of the embedding when all the path-like trees are paths. The arrows indicate the embedded edge is connected to $X=\{v_{r+1},\ldots,v_n\}$.
• Figure 4: Embedding part of $P_i$ and $P_{i+1}$ together to cover $W_{i,j}$ in the four cases $j\leq i-2$, $j=i-1$, $j=i$ and $j=i+1$, while omitting only the vertices marked by a $\times$.
• Figure 5: Embedding part of $P_i$ and $P_{i+1}$ together to cover $W_{i,j}$ when $j\leq i-2$, while omitting only the vertices marked by a $\times$, when the 'artificial end' of $P_i$ used is not a path but a more complicated tree. Here the multiple arrows leading to the right make it more difficult to embed this artificial end, so additional vertices in $W_{i,j}$ may be omitted, to be later covered by other leaves of the tree.

Theorems & Definitions (53)

• Conjecture 1.1: The tree packing conjecture (TPC)
• Theorem 1.2
• Theorem 2.1
• Definition 3.1
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Lemma 3.4
• proof