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

We show that all invertible n × n matrices over any finite field 𝔽_q can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or dividing a row by the generator of the multiplicative group of 𝔽_q. This even holds in the more general setting where the pairs of rows that can be added or subtracted are specified by an arbitrary transition tree that has to satisfy some mild constraints. Moreover, we can prescribe the first and the last matrix if n ≥ 3, or n = 2 and q > 2. In other words, the corresponding flip graph on all invertible n × n matrices over 𝔽_q is Hamilton connected if it is not a cycle. This solves yet another special case of Lovász conjecture on Hamiltonicity of vertex-transitive graphs.

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.

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.

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 quest for good spanners of hypercubes is motivated by their usage as network architectures. Spanners that keep short distances while reducing size or that offer multiple edge-disjoint paths between nodes are favoured. In particular, a 3-spanner guarantees that neighbours in the original graph remain within distance 3 in the spanner. There are known constructions of sparse 3-spanners of hypercubes but they are inefficient for small dimensions. We show two constructions efficient for small dimensions, which improve previously known upper bound on the minimal size of hypercube 3-spanners. We present a genetic algorithm with new hypercube-specific operators and fitness functions addressing the problems of finding a 3-spanner of minimal size and finding multiple edge-disjoint spanners. We evaluate it over a variety of settings and emphasize the most notable ones. For small dimensions the computed results match the new improved constructions this paper presents and for higher dimensions they exceed previously known results.

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.

A queue layout of a graph consists of a linear ordering $\sigma$ of its vertices and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to $\sigma$. We show that the n-dimensional hypercube $Q_n$ has a layout into $n-\lfloor \log_2 n \rfloor$ queues for all $n\ge 1$. On the other hand, for every $\varepsilon>0$, every queue layout of $Q_n$ has more than $(\frac{1}{2}-\varepsilon) n-O(1/\varepsilon)$ queues and, in particular, more than $(n-2)/3$ queues. This improves previously known upper and lower bounds on the minimal number of queues in a queue layout of $Q_n$. For the lower bound we employ a new technique of out-in representations and contractions which may be of independent interest.

Given a set $\pc=\{a_i,b_i\}_{i=1}^m$ of pairs of vertices in a graph G, is there a collection of paths $\{P_i\}_{i=1}^m$ such that $P_i$ connects $a_i$ with $b_i$ and $\{V(P_i)\}_{i=1}^m$ partitions $V(G)$? We study this problem for the graph $Q_n-\ff$ obtained from the n-dimensional hypercube $Q_n$ by removing a set $\ff$ of faulty vertices. We show that an obvious necessary condition for the existence of such a partition is also sufficient provided $2|\pc|+ 3|\ff|\le n-3$. As a corollary, we obtain a similar characterization for the existence of a hamiltonian cycle and a hamiltonian path of $Q_n-\ff$ provided $|\ff|\le(n-5)/3$. On the other hand, if the size of $\ff$ is not limited, the problems are NP-complete.