An $n$-Venn diagram is a diagram in the plane consisting of $n$ simple closed curves that intersect only finitely many times such that each of the $2^n$ possible intersections is represented by a single connected region. An $n$-Venn diagram has at most $2^n-2$ crossings, and if this maximum number of crossings is attained, then only two curves intersect in every crossing. To complement this, Bultena and Ruskey considered $n$-Venn diagrams that minimize the number of crossings, which implies that many curves intersect in every crossing. Specifically, they proved that the total number of crossings in any $n$-Venn diagram is at least $L_n:=\lceil\frac{2^n-2}{n-1}\rceil$, and if this lower bound is attained then essentially all $n$ curves intersect in every crossing. Diagrams achieving this bound are called minimum Venn diagrams, and are known only for $n\leq 7$. Bultena and Ruskey conjectured that they exist for all $n\geq 8$. In this work, we establish an asympototic version of their conjecture. For $n=8$ we construct a diagram with 40 crossings, only 3 more than the lower bound $L_8=37$. Furthermore, for every $n$ of the form $n=2^k$ for some integer $k\geq 4$, we construct an $n$-Venn diagram with at most $(1+\frac{33}{8n})L_n=(1+o(1))L_n$ many crossings. Via a doubling trick this also gives $(n+m)$-Venn diagrams for all $0\leq m

In 1984, Winkler conjectured that every simple Venn diagram with n curves can be extended to a simple Venn diagram with n+1 curves. This conjecture is equivalent to the statement that the dual graph of any simple Venn diagram has a Hamilton cycle. In this work, we construct counterexamples to Winkler’s conjecture for all n ≥ 6. As part of this proof, we computed all 3.430.404 simple Venn diagrams with n = 6 curves (even their number was not previously known), among which we found 72 counterexamples. We also disprove another conjecture about the Hamiltonicity of the arrangement graph of a Venn diagram. Specifically, while working on Winkler’s conjecture, Pruesse and Ruskey proved that this graph has a Hamilton cycle for every simple Venn diagram with n curves, and conjectured that this also holds for non-simple diagrams. We construct counterexamples to this conjecture for all n ≥ 4.

For an arrangement \(\mathcal{H}\) of hyperplanes in \(\mathbb{R}^n\) through the origin, a region is a connected subset of \(\mathbb{R}^n \setminus \mathcal{H}\). The graph of regions \(G(\mathcal{H})\) has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from \(\mathcal{H}\). We aim to compute a Hamiltonian path or cycle in the graph \(G(\mathcal{H})\), i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if \(\mathcal{H}\) is a supersolvable arrangement, then the graph of regions \(G(\mathcal{H})\) has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions \(\mathsf{P}(\mathcal{H}, R_0)\), obtained by orienting the graph \(G(\mathcal{H})\) away from a particular base region \(R_0\). Our second main result is that if \(\mathcal{H}\) is supersolvable and \(R_0\) is a canonical base region, then for any lattice congruence \(\equiv\) on \(\mathsf{P}(\mathcal{H}, R_0) =: L\), the cover graph of the quotient lattice \(L/{\equiv}\) has a Hamiltonian path.

Abstract. The [Formula: see text]-dimensional hypercube graph [Formula: see text] has as vertices all subsets of [Formula: see text], and an edge between any two sets that differ in a single element. The Ruskey–Savage conjecture asserts that every matching of [Formula: see text], [Formula: see text], can be extended to a Hamilton cycle, i.e., to a cycle that visits every vertex exactly once. We prove that every matching of [Formula: see text], [Formula: see text], can be extended to a cycle that visits at least a [Formula: see text]-fraction of all vertices.

An elimination tree for a connected graph \(G\) is a rooted tree on the vertices of \(G\) obtained by choosing a root \(x\) and recursing on the connected components of \(G-x\) to produce the subtrees of \(x\) . Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung–Hoang–Mütze–Williams combinatorial generation framework to elimination trees and prove that all elimination trees for a chordal graph \(G\) can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph \(G\) can be implemented in time \({\mathcal{O}}(\sigma)\) on average per generated elimination tree, where \(\sigma=\sigma(G)\) denotes the maximum number of edges of an induced star in \(G\) . If \(G\) is a tree, we improve this to a loopless algorithm running in time \({\mathcal{O}}(1)\) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of \(G\) , rather than just Hamilton path, if the graph \(G\) is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of \(G\) if and only if \(G\) is chordal.

The associahedron is the graph $\mathcal{G}_N$ that has as nodes all triangulations of a convex $N$-gon, and an edge between any two triangulations that differ in a flip operation. A flip removes an edge shared by two triangles and replaces it by the other diagonal of the resulting 4-gon. In this paper, we consider a large collection of induced subgraphs of $\mathcal{G}_N$ obtained by Ramsey-type colorability properties. Specifically, coloring the points of the $N$-gon red and blue alternatingly, we consider only colorful triangulations, namely triangulations in which every triangle has points in both colors, i.e., monochromatic triangles are forbidden. The resulting induced subgraph of $\mathcal{G}_N$ on colorful triangulations is denoted by $\mathcal{F}_N$. We prove that $\mathcal{F}_N$ has a Hamilton cycle for all $N\geq 8$, resolving a problem raised by Sagan, i.e., all colorful triangulations on $N$ points can be listed so that any two cyclically consecutive triangulations differ in a flip. In fact, we prove that for an arbitrary fixed coloring pattern of the $N$ points with at least 10 changes of color, the resulting subgraph of $\mathcal{G}_N$ on colorful triangulations (for that coloring pattern) admits a Hamilton cycle. We also provide an efficient algorithm for computing a Hamilton path in $\mathcal{F}_N$ that runs in time $\mathcal{O}(1)$ on average per generated node. This algorithm is based on a new and algorithmic construction of a tree rotation Gray code for listing all $n$-vertex $k$-ary trees that runs in time $\mathcal{O}(k)$ on average per generated tree.

Abstract. In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets, and polytopes. Our method is based on a simple and versatile algorithm for computing a Hamilton path on the skeleton of a 0/1-polytope [Formula: see text], where [Formula: see text]. The algorithm uses as a black box any algorithm that solves a variant of the classical linear optimization problem [Formula: see text], and the resulting delay, i.e., the running time per visited vertex on the Hamilton path, is larger than the running time of the optimization algorithm only by a factor of [Formula: see text]. When [Formula: see text] encodes a particular class of combinatorial objects, then traversing the skeleton of the polytope [Formula: see text] along a Hamilton path corresponds to listing the combinatorial objects by local change operations; i.e., we obtain Gray code listings. As concrete results of our general framework, we obtain efficient algorithms for generating all ([Formula: see text]-optimal) bases and independent sets in a matroid; ([Formula: see text]-optimal) spanning trees, forests, matchings, maximum matchings, and [Formula: see text]-optimal matchings in a graph; vertex covers, minimum vertex covers, [Formula: see text]-optimal vertex covers, stable sets, maximum stable sets, and [Formula: see text]-optimal stable sets in a bipartite graph; as well as antichains, maximum antichains, [Formula: see text]-optimal antichains, and [Formula: see text]-optimal ideals of a poset. Specifically, the delay and space required by these algorithms are polynomial in the size of the matroid ground set, graph, or poset, respectively. Furthermore, all of these listings correspond to Hamilton paths on the corresponding combinatorial polytopes, namely the base polytope, matching polytope, vertex cover polytope, stable set polytope, chain polytope, and order polytope, respectively. As another corollary from our framework, we obtain an [Formula: see text] delay algorithm for the vertex enumeration problem on 0/1-polytopes [Formula: see text], where [Formula: see text] and [Formula: see text], and [Formula: see text] is the time needed to solve the linear program [Formula: see text]. This improves upon the 25-year-old [Formula: see text] delay algorithm due to Bussieck and Lübbecke.

In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets and polytopes. Our method relies on a simple and versatile algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope $\operatorname{conv}(X)$, where $X \subseteq\{0,1\}^{n}$. The algorithm uses as a black box any algorithm that solves a variant of the classical linear optimization problem $\min \{w \cdot x \mid x \in X\}$, and the resulting delay, i.e., the running time per visited vertex on the Hamilton path, is only by a factor of $\log n$ larger than the running time of the optimization algorithm. When X encodes a particular class of combinatorial objects, then traversing the skeleton of the polytope $\operatorname{conv}(X)$ along a Hamilton path corresponds to listing the combinatorial objects by local change operations, i.e., we obtain Gray code listings. As concrete results of our general framework, we obtain efficient algorithms for generating all (c-optimal) bases and independent sets in a matroid; (c-optimal) spanning trees, forests, matchings, maximum matchings, and c-optimal matchings in a general graph; vertex covers, minimum vertex covers, c-optimal vertex covers, stable sets, maximum stable sets and c-optimal stable sets in a bipartite graph; as well as antichains, maximum antichains, c-optimal antichains, and c-optimal ideals of a poset. Specifically, the delay and space required by these algorithms are polynomial in the size of the matroid ground set, graph, or poset, respectively. Furthermore, all of these listings correspond to Hamilton paths on the corresponding combinatorial polytopes, namely the base polytope, matching polytope, vertex cover polytope, stable set polytope, chain polytope and order polytope, respectively. As another corollary from our framework, we obtain an $\mathcal{O}\left(t_{\text{LP}} \log n\right)$ delay algorithm for the vertex enumeration problem on 0/1-polytopes $\left\{x \in \mathbb{R}^{n} \mid A x \leq b\right\}$, where $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^{m}$, and $t_{\text{LP}}$ is the time needed to solve the linear program $\min \{w \cdot x \mid A x \leq b\}$. This improves upon the 25-year old $\mathcal{O}\left(t_{\text{LP}} n\right)$ delay algorithm due to Bussieck and Lübbecke.

In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn. Specifically, we propose algorithms for generating different classes of binary trees that are characterized by avoiding one or more of these generalized patterns. This is achieved by applying the recent Hartung-Hoang-Mütze-Williams generation framework, by encoding binary trees via permutations. In particular, we establish a one-to-one correspondence between tree patterns and certain mesh permutation patterns. We also conduct a systematic investigation of all tree patterns on at most 5 vertices, and we establish bijections between pattern-avoiding binary trees and other combinatorial objects, in particular pattern-avoiding lattice paths and set partitions.

In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs (SODA 2022). Secondly, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud (FPSAC 2022). This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group (SODA 2020). Our algorithms are derived from the Hartung-Hoang-Mutze-Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage-Squire-West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs. * Arturo Merino was supported by ANID Becas Chile 2019-72200522. Torsten Mütze was supported by Czech Science Foundation grant GA 22-15272S. Arturo Merino and Torsten Mütze were also supported by German Science Foundation grant 413902284.

For integers ‍k≥ 1 and n≥ 2k+1, the Kneser graph ‍K(n,k) has as vertices all k-element subsets of an n-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph ‍K(5,2). This problem received considerable attention in the literature, including a recent solution for the sparsest case n=2k+1. The main contribution of this paper is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph ‍J(n,k,s) has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly ‍s. Clearly, we have K(n,k)=J(n,k,0), i.e., generalized Johnson graph include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász’ conjecture on Hamilton cycles in vertex-transitive graphs from ‍1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway’s Game of Life, and to analyze this system combinatorially and via linear algebra.

Abstract A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants, including aspect-ratio-universal rectangulations. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework.

Abstract. In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. First, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage–Squire–West construction with a recent algorithm for generating elimination trees of chordal graphs. Second, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud. This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group. Our algorithms are derived from the Hartung–Hoang–Mütze–Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage–Squire–West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.

We say that a Hamilton cycle $C=(x_1,\ldots,x_n)$ in a graph $G$ is $k$-symmetric, if the mapping $x_i\mapsto x_{i+n/k}$ for all $i=1,\ldots,n$, where indices are considered modulo $n$, is an automorphism of $G$. In other words, if we lay out the vertices $x_1,\ldots,x_n$ equidistantly on a circle and draw the edges of $G$ as straight lines, then the drawing of $G$ has $k$-fold rotational symmetry, i.e., all information about the graph is compressed into a $360^\circ/k$ wedge of the drawing. The maximum $k$ for which there exists a $k$-symmetric Hamilton cycle in $G$ is referred to as the Hamilton compression of $G$. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases we determine their Hamilton compression exactly, and in other cases we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

An elimination tree for a connected graph G is a rooted tree on the vertices of G obtained by choosing a root x and recursing on the connected components of G–x to produce the subtrees of x. Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-Mütze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph G can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph G can be implemented in time (m + n) per generated elimination tree, where m and n are the number of edges and vertices of G, respectively. If G is a tree, we improve this to a loopless algorithm running in time (1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of G, rather than just Hamilton path, if the graph G is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of G if and only if G is chordal.

Given integers $k\geq 2$ and $a_1,\ldots,a_k\geq 1$, let $\boldsymbol{a}:=(a_1,\ldots,a_k)$ and $n:=a_1+\cdots+a_k$. An $\boldsymbol{a}$-multiset permutation is a string of length $n$ that contains exactly $a_i$ symbols $i$ for each $i=1,\ldots,k$. In this work we consider the problem of exhaustively generating all $\boldsymbol{a}$-multiset permutations by star transpositions, i.e., in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations ($a_1=\cdots=a_k=1$) can be generated by star transpositions, while combinations ($k=2$) can be generated by these operations if and only if they are balanced ($a_1=a_2$), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter $Δ(\boldsymbol{a}):=n-2\max\{a_1,\ldots,a_k\}$ that allows us to distinguish three different regimes for this problem. We show that if $Δ(\boldsymbol{a})<0$, then a star transposition Gray code for $\boldsymbol{a}$-multiset permutations does not exist. We also construct such Gray codes for the case $Δ(\boldsymbol{a})>0$, assuming that they exist for the case $Δ(\boldsymbol{a})=0$. For the case $Δ(\boldsymbol{a})=0$ we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.

The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length 2(n+1) with exactly n+1 many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book The Art of Computer Programming, Knuth raised a stronger form of this conjecture (Problem 56 in section 7.2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has 2n+1 blocks of the same length, and each block is obtained by adding s=1 (modulo 2n+1) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts $s\geq 1$ that are coprime to 2n+1. We also present an algorithm to compute this ordering, generating each new bitstring in $\mathcal{O}(n)$ time, using $\mathcal{O}(n)$ memory in total.

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework.

The well-known middle levels conjecture asserts that for every integer n ≥ 1, all binary strings of length 2(n + 1) with exactly n + 1 many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book ‘The Art of Computer Programming Vol. 4A’ Knuth raised a stronger form of this conjecture (Problem 56 in Section 7.2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has 2n + 1 blocks of the same length, and each block is obtained by adding s = 1 (modulo 2n + 1) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts s ≥ 1 that are coprime to 2n + 1. We also present an algorithm to compute this ordering, generating each new bitstring in (n) time, using (n) memory in total.

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an n-element set by adjacent transpositions; the binary reflected Gray code to generate all n-bit strings by flipping a single bit in each step; the Gray code for generating all n-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an n-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of patternavoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group Sn. Recently, Pilaud and Santos realized all those lattice congruences as (n – 1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.

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 noncrossing perfect matchings on a set of $n$ points in convex position. In addition, we consider two flip graphs on classes of nongeometric 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 nonexistence of rainbow cycles for different values of $r$, $n$, and $k$.

Abstract Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called $$\alpha$$ α -orientations of a graph G, in which every vertex v has a specified outdegree $$\alpha (v)$$ α ( v ) , and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $$\alpha$$ α -orientations of a planar graph G is at most two is -complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569–576, 2019).

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices into supervertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance $d$, the corresponding supervertices remain at distance at least $\varphi(d)$ in the contracted graph, where $\varphi$ is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $\varphi(x)=x/\alpha-\beta$, where $\alpha\geq 1$ and $\beta\geq 0$ are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. Further we analyze the asymptotic behavior of contractions, and find efficient algorithms to compute (nonoptimal) contractions despite our hardness results.

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. \nIn this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. \nFormally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance d, the corresponding super-vertices remain at distance at least \\varphi(d) in the contracted graph, where \\varphi is a tolerance function bounding the permitted distance distortion. \nWe present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions \\varphi(x)=x/\\alpha-\\beta, where \\alpha \\geq 1 and \\beta \\geq 0 are arbitrary real-valued parameters. \nSpecifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. \nFurther we analyze the asymptotic behavior of the size of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.

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.

For any integer n ≥ 1, a middle levels Gray code is a cyclic listing of all bitstrings of length 2 n +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 ≥ 1 has been the subject of intensive research during the past 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 ℓ bitstrings in the Gray code in time O ( n ℓ (1+ n /ℓ)), which is O ( n ) on average per bitstring provided that ℓ = Ω ( n ).

We consider the algorithmic problem of generating each subset of [ n ] : = { 1 , 2 , … , n } whose size is in some interval [ k , l ] , 0 ≤ k ≤ l ≤ 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 ( 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 O ( 1 ) time per generated set and 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 2 k + 1 induced by all levels [ k − c , k + 1 + c ] , c ∈ { 0 , 1 , … , 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 ≥ 1 a middle levels Gray code is a cyclic listing of all n-element and (n + 1)- element subsets of {1,2,…, 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 ≥ 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. To appear in Proc. London Math. Soc., 2014]. In a follow-up paper [T. Mütze and J. Nummenpalo. An efficient algorithm for computing a middle levels Gray code. Proc. ESA, 2015] this existence proof was turned into an algorithm that computes each new set in the Gray code in time O(n) on average. In this work we complete this line of research by presenting an algorithm for computing a middle levels Gray code in optimal time and space: Each new set is generated in time O(1), and the required space is O(n).

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 nonedges 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 chosen not 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 these general results with several examples. The technique proposed here has been used by Balogh and Butterfield [Discrete Math., 310 (2010), pp. 3653--3657] to derive the first nontrivial upper bounds for the threshold of the game where $F$ is a triangle and more than two colors are available.

Consider the following generalized notion of graph coloring: a coloring of the vertices of a graph G is valid w.r.t. some given graph F if there is no copy of F in G whose vertices all receive the same color. We study the problem of computing valid colorings of the binomial random graph Gn,p on n vertices with edge probability p = p(n) in the following online setting: the vertices of an initially hidden instance of Gn,p are revealed one by one (together with all edges leading to previously revealed vertices) and have to be colored immediately and irrevocably with one of r available colors. It is known that for any fixed graph F and any fixed integer r ≥ 2 this problem has a threshold p0 (F, r, n) in the following sense: For any function p(n) = o(p0) there is a strategy that a.a.s. (asymptotically almost surely, i.e., with probability tending to 1 as n tends to infinity) finds an r-coloring of Gn,p that is valid w.r.t. F online, and for any function p(n) = ω(p0) any online strategy will a.a.s. fail to do so. In this work we establish a general correspondence between this probabilistic problem and a 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. This characterization allows us to compute, for any F and r, a value γ = γ(F, r) such that the threshold of the probabilistic problem is given by p0(F, r, n) = n−γ. Our approach yields polynomial-time coloring algorithms that a.a.s. find valid colorings of Gn,p online in the entire regime below the respective thresholds, i.e., for any p(n) = o(n−γ).

In an effort to better understand connectivity and capacity in wireless networks, the log-normal shadowing radio propagation model is used to capture radio irregularities and obstacles in the transmission path. Existing results indicate that log-normal shadowing results in higher connectivity and interference levels as shadowing (i.e., the radio irregularity) increases. In this paper we demonstrate that such a behavior is mainly caused by an unnatural bias of the log-normal shadowing radio propagation model that results in a larger transmission range as shadowing increases. To avoid this effect, we analyze connectivity and interference under log-normal shadowing using a normalization that compensates for the enlarged radio transmission range. Our analysis shows that log-normal shadowing still improves the connectivity of a wireless network and even reduces interference. We explain this behavior by studying in detail what network parameters are affected by shadowing. Our results indicate that, when it comes to connectivity and interference, an analysis based on a circular transmission range leads to worst case results.