Constructions of Snake-in-the-Box Codes for Rank Modulation

Snake-in-the-box code is a Gray code which is capable of detecting a single error. Gray codes are important in the context of the rank modulation scheme which was suggested recently for representing information in flash memories. For a Gray code in this scheme the codewords are permutations, two consecutive codewords are obtained by using the "push-to-the-top" operation, and the distance measure is defined on permutations. In this paper the Kendall's $\tau$-metric is used as the distance measure. We present a general method for constructing such Gray codes. We apply the method recursively to obtain a snake of length $M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}$ for permutations of $S_{2n+1}$, from a snake of length $M_{2n-1}$ for permutations of~$S_{2n-1}$. Thus, we have $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4338$, improving on the previous known ratio of $\lim\limits_{n\to \infty} \frac{1}{\sqrt{\pi n}}$. By using the general method we also present a direct construction. This direct construction is based on necklaces and it might yield snakes of length $\frac{(2n+1)!}{2} -2n+1$ for permutations of $S_{2n+1}$. The direct construction was applied successfully for $S_7$ and $S_9$, and hence $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4743$.Comment: IEEE Transactions on Information Theor

Sparse Kneser graphs are Hamiltonian

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k\geq 3$ and $a\geq 0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k\geq 6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words