Seven largest trees pack

Abstract

The Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees $T_2,\dots,T_{n-1}, T_n$ such that $T_i$ has $i$ vertices pack into $K_n$. The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that $k$ largest trees pack. The latter is true if none tree is a star, but in general, it is known only for $k=5$. In this paper we prove, among other results, that seven largest trees packThe Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees $T_2,\dots,T_{n-1}, T_n$ such that $T_i$ has $i$ vertices pack into $K_n$. The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that $k$ largest trees pack. The latter is true if none tree is a star, but in general, it is known only for $k=5$. In this paper we prove, among other results, that seven largest trees pack