arXiv subjects
Publications and source records attributed to Torsten Mütze.
It is shown that shift graphs can be realized as disjointness graphs of 1-intersecting curves in the plane. This implies that the latter class of graphs is not $χ$-bounded.
The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon, and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly $r$ times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of $r$-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex $n$-gon, the flip graph of plane trees on an arbitrary set of $n$ points, and the flip graph of non-crossing perfect matchings on a set of $n$ points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of $\{1,2,\dots,n\}$ and the flip graph of $k$-element subsets of $\{1,2,\dots,n\}$. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of $r$, $n$ and~$k$.
We consider the algorithmic problem of generating each subset of $[n]:=\{1,2,\ldots,n\}$ whose size is in some interval $[k,l]$, $0\leq k\leq l\leq n$, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For $k=0$ and $l=n$ this is the classical problem of generating all $2^n$ subsets of $[n]$ by element additions/removals, and for $k=l$ this is the classical problem of generating all $\binom{n}{k}$ subsets of $[n]$ by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values $n$, $k$, and $l$, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant $\mathcal{O}(1)$ time per generated set and $\mathcal{O}(n)$ space. Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the $n$-dimensional cube $Q_n$ induced by all levels $[k,l]$. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of $Q_{2k+1}$ induced by all levels $[k-c,k+1+c]$, $c\in\{0,1,\ldots,k\}$, has a Hamilton cycle. We also prove an approximate version of this generalized conjecture, showing that this graph has a cycle that visits a $(1-o(1))$-fraction of all vertices.
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. Mütze. 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 $\ell=Ω(n)$.
A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer.
A Dyck path with $2k$ steps and $e$ flaws is a path in the integer lattice that starts at the origin and consists of $k$ many $\nearrow$-steps and $k$ many $\searrow$-steps that change the current coordinate by $(1,1)$ or $(1,-1)$, respectively, and that has exactly $e$ many $\searrow$-steps below the line $y=0$. Denoting by $D_{2k}^e$ the set of Dyck paths with $2k$ steps and $e$ flaws, the Chung-Feller theorem asserts that the sets $D_{2k}^0,D_{2k}^1,\ldots,D_{2k}^k$ all have the same cardinality $\frac{1}{k+1}\binom{2k}{k}=C_k$, the $k$-th Catalan number. The standard combinatorial proof of this classical result establishes a bijection $f'$ between $D_{2k}^e$ and $D_{2k}^{e+1}$ that swaps certain parts of the given Dyck path $x$, with the effect that $x$ and $f'(x)$ may differ in many positions. In this paper we strengthen the Chung-Feller theorem by presenting a simple bijection $f$ between $D_{2k}^e$ and $D_{2k}^{e+1}$ which has the additional feature that $x$ and $f(x)$ differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of $f$ in constant time per generated Dyck path. As an application, we use our minimum-change bijection $f$ to construct cycle-factors in the odd graph $O_{2k+1}$ and the middle levels graph $M_{2k+1}$ --- two intensively studied families of vertex-transitive graphs --- that consist of $C_k$ many cycles of the same length.
For integers $k\geq 1$ and $n\geq 2k+1$ the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as vertices all $k$-element and $(n-k)$-element subsets of $[n]$ and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph $K(5,2)$ have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs $H(n,k)$. We also establish the existence of cycles that visit almost all vertices in Kneser graphs $K(n,k)$ when $n=2k+o(k)$, generalizing and improving upon previous results on this problem.
Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. The middle levels conjecture asserts that this graph has a Hamilton cycle for every $n\geq 1$. This conjecture originated probably with Havel, Buck and Wiedemann, but has also been attributed to Dejter, Erdős, Trotter and various others, and despite considerable efforts it remained open during the last 30 years. In this paper we prove the middle levels conjecture. In fact, we construct $2^{2^{Ω(n)}}$ different Hamilton cycles in the middle layer graph, which is best possible.
Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. In this work we present an inductive construction of a large family of 2-factors in the middle layer graph for all $n\geq 1$. We also investigate how the choice of certain parameters used in the construction affects the number and lengths of the cycles in the resulting 2-factor.
We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{ö}dl and Ruci{ń}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available.
Consider the following random process: The vertices of a binomial random graph $G_{n,p}$ are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number $r$ of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph $F$ in the process. Our first main result is that for any $F$ and $r$, the threshold function for this problem is given by $p_0(F,r,n)=n^{-1/m_1^*(F,r)}$, where $m_1^*(F,r)$ denotes the so-called \emph{online vertex-Ramsey density} of $F$ and $r$. This parameter is defined via a purely deterministic two-player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any $F$ and $r$, the online vertex-Ramsey density $m_1^*(F,r)$ is a computable rational number. Our lower bound proof is algorithmic, i.e., we obtain polynomial-time online algorithms that succeed in coloring $G_{n,p}$ as desired with probability $1-o(1)$ for any $p(n) = o(n^{-1/m_1^*(F,r)})$.
For any graph $F$ and any integer $r\geq 2$, the \emph{online vertex-Ramsey density of $F$ and $r$}, denoted $m^*(F,r)$, is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs.\ Builder). This parameter was introduced in a recent paper \cite{mrs11}, where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs.\ the binomial random graph $G_{n,p}$). For a large class of graphs $F$, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for $m^*(F,r)$ are known. In this work we show that for the case where $F=P_\ell$ is a (long) path, the picture is very different. It is not hard to see that $m^*(P_\ell,r)= 1-1/k^*(P_\ell,r)$ for an appropriately defined integer $k^*(P_\ell,r)$, and that the greedy strategy gives a lower bound of $k^*(P_\ell,r)\geq \ell^r$. We construct and analyze Painter strategies that improve on this greedy lower bound by a factor polynomial in $\ell$, and we show that no superpolynomial improvement is possible.
Consider the following probabilistic one-player game: The board is a graph with $n$ vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all non-edges and is presented to the player, henceforth called Painter. Painter must assign one of $r$ available colors to each edge immediately, where $r \geq 2$ is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph $F$ has been created, and Painter's goal is to 'survive' for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are not chosen randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number $d$, the ratio of edges to vertices in all subgraphs of the evolving board never exceeds $d$. We show that the existence of a winning strategy for Builder in this deterministic game implies an upper bound of $n^{2-1/d}$ for the threshold of the original probabilistic game. Moreover, we show that the best bound that can be derived in this way is indeed the threshold of the game if $F$ is a forest. We illustrate our technique with several examples, and derive new explicit bounds for the case when $F$ is a path.