A Counterexample to a Conjecture on Packing Two Copies of a Tree into its Third Power

Abstract

A graph \(H\) of order \(n\) is said to be embeddable in a graph \(G\) of order \(n\), if \(G\) contains a spanning subgraph isomorphic to \(H\). It is well known that any non-star tree \(T\) of order \(n\) is embeddable in its complement (i.e. in \(K_n – E(T)\)). In the paper “Packing two copies of a tree into its fourth power” by Hamamache Kheddouci, Jean-Francois Saclé, and Mariusz Wodgniak, Discrete Mathematics 213 (2000), 169-178, it is proved that any non-star tree \(T\) is embeddable in \(T^4 – E(T)\). They asked whether every non-star tree \(T\) is embeddable in \(T^3 – E(T)\). In this paper, answering their question negatively, we show that there exist trees \(T\) such that \(T\) is not embeddable in \(T^3 – E(T)\).