A Note on Packing Complete Graphs with Trees

Gyárfás and Lehel conjectured that any collection of trees \(T_2, T_3, \ldots, T_n\) on \(2, 3, \ldots, n\) vertices respectively, can be packed into the complete graph on \(n\) vertices. Fishburn proved that the conjecture is true for some classes of trees and for all trees up to \(n = 9\). Pritikin characterized the trees for which Fishburn’s proof works and extended the classes of trees for which the conjecture is known to be true. Using a computer, we have shown that the conjecture is true through \(n = 11\), but also that an approach suggested by Fishburn is unlikely to work in general.