The domination number of exchanged hypercubes

Exchanged hypercubes [Loh et al., IEEE Transactions on Parallel and Distributed Systems 16 (2005) 866–874] are spanning subgraphs of hypercubes with about one half of their edges but still with many desirable properties of hypercubes. Lower and upper bounds on the domination number of exchanged hypercubes are proved which in particular imply that γ(EH(2, t)) = 2 holds for any t ≥ 2. Using Hamming codes we also prove that γ(EH(s,2−1)) ≤ (2s−2k)γ(Qt)+2t(γ(Q−s)+1) holds for s ≥ k ≥ 3.