Perfect snake-in-the-box codes for rank modulation

TL;DR

This paper proves the Hamiltonicity of a specific directed Cayley graph related to permutations for all odd n except 5, solving a problem in flash memory rank modulation and disproving a prior conjecture.

Contribution

It establishes Hamiltonicity for the Cayley graph generated by certain permutations for all odd n≠5, advancing understanding in rank modulation coding schemes.

Findings

Hamiltonicity proven for all odd n≠5

Disproves a conjecture of Horovitz and Etzion

Supports a conjecture of Yehezkeally and Schwartz

Abstract

For odd n, the alternating group on n elements is generated by the permutations that jump an element from any odd position to position 1. We prove Hamiltonicity of the associated directed Cayley graph for all odd n not equal to 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.