Hamiltonian cycles in hypercubes with 2n-42n-4 faulty edges

In this paper we consider Hamiltonian cycles in hypercubes with faulty edges and present a simple proof that the hypercube QnQn is not Hamiltonian if it contains a subgraph T such that: (i) exactly half of the vertices in T have parity 0 (the other half have parity 1) and (ii) all edges joining the nodes of parity 0 (or 1) with nodes outside T, are faulty. We will call such a subgraph a trap disconnected halfway  . Moreover, we show that for n⩾5n⩾5, the cube QnQn with 2n-42n-4 faulty edges is not Hamiltonian only if it contains a trap T disconnected halfway with T being a subcube of dimension 1 or 2.