A short proof of the middle levels theorem

Consider the graph that has as vertices all bitstrings of length $2n+1$ with exactly $n$ or $n+1$ entries equal to 1, and an edge between any two bitstrings that differ in exactly one bit. The well-known middle levels conjecture asserts that this graph has a Hamilton cycle for any $n\geq 1$. In this paper we present a new proof of this conjecture, which is much shorter and more accessible than the original proof

A constant-time algorithm for middle levels Gray codes

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive subsets differ in adding or removing a single element. The question whether such a Gray code exists for any $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T. M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence proof was turned into an algorithm that computes each new set in the Gray code in time $\mathcal{O}(n)$ on average. In this work we present an algorithm for computing a middle levels Gray code in optimal time and space: each new set is generated in time $\mathcal{O}(1)$ on average, and the required space is $\mathcal{O}(n)$

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

A constant-time algorithm for middle levels Gray codes

For any integer~$n\geq 1$, a \emph{middle levels Gray code} is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive sets differ in adding or removing a single element. The question whether such a Gray code exists for any~$n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T.~M\"utze. Proof of the middle levels conjecture. \textit{Proc. London Math. Soc.}, 112(4):677--713, 2016]. In a follow-up paper [T.~M\"utze and J.~Nummenpalo. An efficient algorithm for computing a middle levels Gray code. \textit{ACM Trans. Algorithms}, 14(2):29~pp., 2018] this existence proof was turned into an algorithm that computes each new set in the Gray code in time~\cO(n) on average. In this work we present an algorithm for computing a middle levels Gray code in optimal time and space: each new set is generated in time~\cO(1), and the required space is~\cO(n)

Sparse Kneser graphs are Hamiltonian

For integers k≥1 and n≥2k+1, the Kneser graph K(n,k) is the graph whose vertices are the k-element subsets of {1,…,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≥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+2a,k) with k≥3 and a≥0 have a Hamilton cycle. We also prove that K(2k+1,k) has at least 22k−6 distinct Hamilton cycles for k≥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

Efficient computation of middle levels Gray codes

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all bitstrings of length $2n+1$ that have either $n$ or $n+1$ entries equal to 1 such that any two consecutive bitstrings in the list differ in exactly one bit. The question whether such a Gray code exists for every $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In this work we provide the first efficient algorithm to compute a middle levels Gray code. For a given bitstring, our algorithm computes the next $\ell$ bitstrings in the Gray code in time $\mathcal{O}(n\ell(1+\frac{n}{\ell}))$, which is $\mathcal{O}(n)$ on average per bitstring provided that $\ell=\Omega(n)$

Random lattice walks in a Weyl chamber of type A or B and non-intersecting lattice paths

Die vorliegende Arbeit beschäftigt sich mit zwei eng verwandten Modellen: Gitterpfaden in einer Weylkammer vom Typ B und nichtüberschneidenden Gitterpfaden im ganzzahligen Gitter aufgespannt durch die Vektoren {(1,1),(1,-1)} mit Schritten aus dieser Menge. Diese Gitterpfadmodelle sind von zentraler Bedeutung z.B. in der Kombinatorik und der statistischen Mechanik. In der statistischen Mechanik dienen diese Modelle der Beschreibung bestimmter nicht-kollidierender Teilchen-Systeme. Die Bedeutung von Gitterpfadmodellen in der Kombinatorik ist teilweise begründet durch ihre interessanten kombinatorischen Eigenschaften, vor allem aber auch durch die engen Beziehungen zu zahlreichen zentralen kombinatorischen Objekten wie z.B. Integer Partitions, Plane Partitions und Young Tableaux. Im ersten Teil dieser Arbeit werden asymptotische Formeln für die Anzahl von Gitterpfaden in einer Weylkammer vom Typ B für eine allgemeine Klasse von Schritten hergeleitet. Die Klasse der zulässigen Schritte wird hierbei durch die Forderung der "Reflektierbarkeit" der resultierenden Pfade beschränkt. Spezialfälle dieser asymptotischen Formel lösen in der Literatur aufgeworfene Probleme und liefern bekannte Resultate für zweidimensionale Vicious Walkers Modelle und sogenannte k-non-crossing tangled diagrams. Im zweiten Teil werden die Zufallsvariablen "Höhe" und "Ausdehnung" auf der Menge aller nichtüberschneidenden Gitterpfade mit n Schritten sowie auf der Teilmenge all jener auf die obere Halbebene beschränkten nichtüberschneidenden Gitterpfade mit n Schritten studiert. Unter der Annahme einer Gleichverteilung auf diesen Mengen wird die asymptotische Verteilung beider Zufallsvariablen bestimmt. Weiters werden die ersten beiden Terme der asymptotischen Entwicklung aller Momente der Zufallsvariable "Höhe" ermittelt. Dies löst ein in der Literatur aufgeworfenes Problem, und verallgemeinert ein bekanntes Resultat über die Höhe ebener Wurzelbäume.This thesis is concerned with two closely related lattice walk models: lattice walks in a Weyl chamber type B and non-intersecting lattice paths on the integer lattice spanned by the vectors {(1,1),(1,-1)} with steps from this set. These models play an important role in, e.g., combinatorics and statistical mechanics. In statistical mechanics, non-intersecting lattice paths serve as models for certain non-colliding particle systems. From a combinatorial point of view, lattice paths models are very natural objects to study, partly because of their intrinsic interesting combinatorics, and partly because of their close relationship to many other important combinatorial structures, such as integer partitions, plane partitions and Young tableaux. In the first part of this thesis, we determine asymptotics for the number of lattice walks in a Weyl chamber of type B for a general class of steps. The class of admissible steps is determined by requiring the walks to be "reflectable". As special cases, these asymptotics include several results found in the literature, e.g., asymptotics for certain vicious walkers models and k-non-crossing tangled diagrams. In the second part of this thesis we study the random variables "height" and "range" on the set of non-intersecting lattice paths of length n as well as on the subset of those non-intersecting lattice paths of length n that are confined to the upper half plane. Assuming the uniform probability distribution on these sets, we determine the asymptotic distribution of both random variables as the number of steps tends to infinity as well as first and second order asymptotics for all moments of the random variable "height". This solves a problem raised in the literature, and generalises a well-known result on the height of random planted plane trees