Abstract
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.
Highlights
The question whether a given graph has a Hamilton cycle is one of the oldest and most fundamental problems in graph theory and computer science, shown to be NP-complete in Karp’s seminal paper [Kar72]
We focus on a well-known instance of this phenomenon—the so-called Kneser graphs
We prove that the odd graphs Ok = K(2k + 1, k) contain Hamilton cycles
Summary
The question whether a given graph has a Hamilton cycle is one of the oldest and most fundamental problems in graph theory and computer science, shown to be NP-complete in Karp’s seminal paper [Kar72]. A Hamilton cycle in the odd graph corresponds to a Gray code listing of all bitstrings of length 2k + 1 with exactly k many 1-bits, such that consecutive bitstrings differ in all but one position It remains open whether our proof can be translated into a constant-time algorithm to generate this Gray code, that is, an algorithm that in each step computes the bit that is not flipped in constant time, using only O(k) memory space and polynomial initialization time. Every spanning tree in Hk corresponds to a collection of flipping cycles such that the symmetric difference of their edge sets and the edges of the cycles in Ck results in a Hamilton cycle in the odd graph Ok. The proof of Theorem 3 exploits the degrees of freedom that are inherent in the construction above to provide double-exponentially many distinct spanning trees in Hk, which give rise to doubleexponentially many distinct Hamilton cycles in Ok. 1.7. The proofs of some technical lemmas formulated in that section are deferred to Sections 4 and 5
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call