A long standing conjecture of Havel (1984)  states that every equipartite tree with maximum degree 3 on 2n vertices is a spanning subgraph of the n-dimensional hypercube. The conjecture is known to be true for many subclasses of trees. Havel and Liebl (1986)  showed that every equipartite caterpillar with maximum degree 3 and having 2n vertices is a spanning subgraph of the n-dimensional hypercube. Subsequently, Havel (1990)  remarked that the problem of verification of the conjecture for subdivisions of caterpillars with maximum degree 3 has remained open. In this paper, we show that a subdivision of a caterpillar with 2n vertices and maximum degree 3 is a spanning subgraph of the n-dimensional hypercube if it is equipartite and has at most n-3 vertices on the spine. The problem of embedding such trees that have spines of arbitrary length is still open. © 2011 Elsevier B.V. All rights reserved.
Choudum, S. A., & Lavanya, S. (2011). Embedding a subclass of trees into hypercubes. Discrete Mathematics, 311(10–11), 866–871. https://doi.org/10.1016/j.disc.2011.02.011