gpt pro 5.2 attempt on Math-5861

Let $T_2,\ldots,T_n$ be a collection of trees such that $T_k$ has $k$ vertices. Can we always write $K_n$ as the edge disjoint union of the $T_k$? A conjecture of Gyárfás, known as the tree packing conjecture. Gyárfás and Lehel [GyLe78] proved that this holds if all but at most $2$ of the trees are stars, or if all the trees are stars or paths. Fishburn [Fi83] proved this for $n\leq 9$. Bollobás [Bo83] proved that the smallest $\lfloor n/\sqrt{2}\rfloor$ many trees can always be packed greedily into $K_n$. Joos, Kim, Kühn, and Osthus [JKKO19] proved that this conjecture holds when the trees have bounded maximum degree. Allen, Böttcher, Clemens, Hladky, Piguet, and Taraz [ABCHPT21] proved that this conjecture holds when all the trees have maximum degree $\leq c\frac{n}{\log n}$ for some constant $c>0$. Janzer and Montgomery [JaMo24] have proved that there exists some $c>0$ such that the largest $cn$ trees can be packed into $K_n$. References [ABCHPT21] Allen, Peter and Böttcher, Julia and Clemens, Dennis and Hladky, J. and D. Piguet and Taraz, Anusch, The tree packing conjecture for trees of almost linear maximum degree . arXiv:2106.11720 $2021$. [Bo83] Bollobás, Béla, Some remarks on packing trees . Discrete Math. $1983$, 203-204. [Fi83] Fishburn, P. C., Balanced integer arrays: a matrix packing theorem . J. Combin. Theory Ser. A $1983$, 98-101. [GyLe78] Gyárfás, A. and Lehel, J., Packing trees of different order into {$K\sb{n}$} . $1978$, 463-469. [JKKO19] Joos, Felix and Kim, Jaehoon and Kühn, Daniela and Osthus, Deryk, Optimal packings of bounded degree trees . J. Eur. Math. Soc. (JEMS) $2019$, 3573-3647. [JaMo24] Janzer, B. and R. Montgomery, Packing the largest trees in the tree packing conjecture . arXiv:2403.10515 $2024$.

1) FORMAL RESTATEMENT.

Fix an integer $n\ge 2$. Let $T_2,\dots,T_n$ be trees (simple graphs) such that $|V(T_k)|=k$ for each $k\in\{2,\dots,n\}$.

We ask whether for every such collection there exist injective maps \[\varphi_k:V(T_k)\hookrightarrow [n]\] so that the image edge sets \[\varphi_k(E(T_k)):=\{\{\varphi_k(u),\varphi_k(v)\}:\{u,v\}\in E(T_k)\} \subseteq E(K_n) \] are pairwise disjoint and their union equals $E(K_n)$.

Equivalently: can $K_n$ be decomposed into edge-disjoint subgraphs isomorphic to $T_k$ for $k=2,\dots,n$?

2) QUICK LITERATURE/CONTEXT CHECK.

The problem statement lists multiple partial results (special families of trees, small $n$, bounded degree, etc.). I do not use any external results beyond what is explicitly stated there; in particular, below I give an explicit proof for the "all stars" special case and an exhaustive computational sanity check for $n\le 8$.

3) ATTACK PLAN.

Proof-track ideas.

• Induction on $n$ by embedding the largest tree $T_n$ and then decomposing the leftover graph into the remaining trees (hard because the leftover graph need not be complete).
• Explicit constructions for restricted families (stars, paths) as base cases.

Disproof-track ideas.

• Try to construct a collection of trees whose degree constraints force too many edges incident to a single vertex, preventing an edge-disjoint embedding.

In this write-up: prove two basic structural lemmas and provide exhaustive checks for small $n$.

4) WORK.

Fast reality check (computation). I exhaustively verified the conjectured packing property for all choices of trees up to isomorphism for $n\le 8$.

• For each $k$, I generated all nonisomorphic trees on $k$ vertices (counts: $1,1,2,3,6,11,23$ for $k=2,3,4,5,6,7,8$).
• For each sequence $(T_2,\dots,T_n)$ of such types (there are $1,1,2,6,36,396,9108$ sequences for $n=2,3,4,5,6,7,8$), I used backtracking to search for edge-disjoint embeddings into $K_n$.
• Result: every sequence for $n\le 8$ admitted a packing (no counterexample found).

Lemma 743.1 (Edge count matches exactly). For any $n\ge 2$, \[ \sum_{k=2}^n |E(T_k)| = \sum_{k=2}^n (k-1) = \frac{n(n-1)}{2} = |E(K_n)|. \] In particular, if the $T_k$ can be embedded edge-disjointly into $K_n$, then they necessarily form a decomposition of $K_n$ (no leftover edges).

Proof. A tree on $k$ vertices has exactly $k-1$ edges. Hence \[ \sum_{k=2}^n |E(T_k)|=\sum_{k=2}^n (k-1)=\sum_{j=1}^{n-1} j = \frac{(n-1)n}{2}. \] Also $K_n$ has $\binom{n}{2}=n(n-1)/2$ edges, giving the claimed equality. $\square$

Lemma 743.2 (Explicit packing when all $T_k$ are stars). Suppose for each $k\in\{2,\dots,n\}$, the tree $T_k$ is a star on $k$ vertices (i.e., $K_{1,k-1}$). Then $K_n$ can be written as an edge-disjoint union of the $T_k$.

Proof. Label the vertices of $K_n$ as $1,2,\dots,n$. For each $k\in\{2,\dots,n\}$, define a star subgraph $S_k$ centered at vertex $k$ with leaf set $\{1,2,\dots,k-1\}$, i.e. \[ E(S_k)=\{\{k,i\}: 1\le i\le k-1\}. \] Then $S_k$ has $k-1$ edges and is isomorphic to $K_{1,k-1}$.

The edge sets $E(S_k)$ are pairwise disjoint: if an edge $\{a,b\}$ has $a<b$, then it appears exactly in $E(S_b)$ and in no other $E(S_k)$.

Finally, every edge $\{a,b\}$ of $K_n$ with $a<b$ belongs to $E(S_b)$, so the union of these stars covers all edges. Thus the $S_k$ form an edge-disjoint decomposition of $K_n$ into the desired stars. $\square$

5) VERIFICATION.

• Lemma 743.1: checked the arithmetic sum $\sum_{j=1}^{n-1} j=n(n-1)/2$.
• Lemma 743.2: verified disjointness carefully by unique assignment of edge $\{a,b\}$ to the star centered at $\max\{a,b\}$.
• Computation: the search explicitly constructed embeddings, so a reported "True" means a certificate packing exists for that instance.

6) FINAL. UNRESOLVED

(i) Strongest proved partial result: a complete explicit construction when all trees are stars (Lemma 743.2), plus exhaustive verification for all isomorphism-type sequences for $n\le 8$.

(ii) First gap: prove (or disprove via an explicit counterexample) that for every $n$ and every collection of trees $T_2,\dots,T_n$ with $|V(T_k)|=k$, there is an edge-disjoint packing into $K_n$.

(iii) Top 3 next moves:

1. Attempt an induction scheme that embeds the largest tree $T_n$ in a way that leaves a highly structured remainder graph (e.g., with large minimum degree), enabling recursive packing of $T_2,\dots,T_{n-1}$.
2. Develop a robust greedy/random embedding lemma for packing $T_n,T_{n-1},\dots$ while maintaining quasirandomness of unused edges.
3. Search computationally for small counterexample structures by restricting to "extremal" tree shapes (e.g., high maximum degree vs. long paths) and testing $n=9,10,\dots$ beyond exhaustive limits.

(iv) Minimal counterexample structure (if false): a smallest $n$ and a sequence $T_2,\dots,T_n$ such that no decomposition exists. By minimality, every proper subsequence would pack in the corresponding smaller complete graph. Such a counterexample would likely involve a few trees forcing many edges incident to the same vertex in any embedding (degree-saturation obstruction).