A note on the cubical dimension of new classes of binary trees
Résumé
The cubical dimension of a graph G is the smallest dimension of a hypercube into which G is embeddable as a subgraph. The conjecture of Havel [On hamiltonian circuits and spanning trees of hypercubes. ˇ Casopis ˇ Pest. Mat 109 (2) (1984) 135-152] claims that the cubical dimension of every balanced binary tree with 2n vertices, n 1, is n. The 2-rooted complete binary tree of depth n is obtained from two copies of the complete binary tree of depth n by adding an edge linking their respective roots. In this paper, we determine the cubical dimen- sion of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.