# Embedding the Gray Graph in a hypercube

The Gray Graph is the subgraph of a nine-dimensional hypercube induced by the bitvectors that have the form {000,001,010,100}^{3} and that have at least two nonzero bits. I was excited this afternoon thinking that this embedding would give a second nonplanar cubic partial cube (the only known one being the Desargues Graph), but unfortunately it doesn't: e.g. 001001000 and 001010000 are at distance four in the graph via a path 001001000-001001001-001000001-001010001-001010000, but they have Hamming distance only two. In fact the graph is not a partial cube for any hypercube embedding. I guess that means it does provide another natural example of a hypercube subgraph that's induced but not isometric...