Spanning Regular Caterpillars in Hypercubes

If a regular caterpillar is a spanning subgraph of a hypercube, then it has 2n-1legs for somenand its length is 2mfor somem.We prove the converse statement: for everynthere existsm0such that for everym≥m0a regular caterpillar of length 2mwith 2n-1 legs is a spanning subgraph of a hypercube of dimensionn+m.