Let $T_2,\dots ,T_n$ 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 $T_k$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

The edge count obstruction disappears: each tree (T_k) has $k-1$ edges, so [ \sum_{k=2}^n |E(T_k)|=\sum_{k=2}^n (k-1)=\binom{n}{2}=|E(K_n)|. ] So your question is exactly whether one can **pack** the trees (T_2,\dots,T_n) edge-disjointly into (K_n); if that packing exists, it automatically uses *all* edges of (K_n).

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.