Perfect Snake-in-the-Box Codes for Rank Modulation

Abstract

For odd $n$ , the alternating group on $n$ elements is generated by the permutations that jump an element from an odd position to position 1. We prove Hamiltonicity of the associated directed Cayley graph for all odd $n\neq 5$ (a result of Rankin implies that the graph is not Hamiltonian for $n=5$ ). This solves a problem arising in rank modulation schemes for flash memory. Our result disproves a conjecture of Horovitz and Etzion, and proves another conjecture of Yehezkeally and Schwartz.