Some Small Sized Spanning Subgraphs of a Hypercube

The integral of a tree T is the tree obtained by joining one new leaf to each node of T. The broadcast tree Bn is the nth iterated integral of the graph K1 consisting of just one node. We derive the number of embeddings of Bn in the hypercube Qn. We also determine the minimum diameter of a spanning tree of Qn. A special spanning subgraph Rn of Qn is then constructed by suitably joining two broadcast trees Bn and Bn−1. This cubical graph Rn of diameter n serves to prove that asymptotically almost all the edges of Qn can be removed and still the remaining spanning subgraph has diameter n. Other properties of this new family of graphs are investigated.