Hamiltonian cycles in unitary prefix transposition rearrangement graphs

Cayley graphs have been extensively studied by graph and group theorists, computer scientists, molecular biologists and coding theorists. We focus on two challenging problems on Cayley graphs arising on sequence comparison: hamiltonian cycle and graph diameter. A unitary prefix transposition exchanges two adjacent blocks in a permutation such that one block contains the first elements and one of the blocks is unitary. The Unitary Prefix Transposition Rearrangement Graph has the permutations in the Symmetric Group SnSn as its vertex set and two vertices are adjacent if there exists a unitary prefix transposition that applied to a permutation produces the other one. We show that this Cayley graph has diameter n−1n−1 and is hamiltonian.