<p><b>Abstract</b>—In this paper, we first show that the <it>degree four Cayley graph</it> proposed in a paper appearing in the January 1996 issue of <it>IEEE Transactions on Parallel and Distributed Systems</it> is indeed isomorphic to the <it>wrapped butterfly</it>. The isomorphism was first reported by Muga and Wei in the proceedings of PDPTA "96.The isomorphism is shown by using an edge-preserving bijective mapping. Due to the isomorphism, algorithms for the degree four Cayley graph can be easily developed in terms of wrapped butterfly and topological properties of one network can be easily derived in terms of the other. Next, we present the first optimal oblivious one-to-one permutation routing scheme for these networks in terms of the wrapped butterfly. Our algorithm runs in time <tmath>$O(\sqrt{N})$</tmath>, where <tmath>$N$</tmath> is the network size.</p>