An Upper bound on the number of Steiner triple systems

Let STS(n) denote the number of Steiner triple systems on n vertices, and let F(n) denote the number of 1-factorizations of the complete graph on n vertices. We prove the following upper bound. STS(n) <= ((1 + o(1)) (n/e^2))^(n^2/6) F(n) <= ((1 + o(1)) (n/e^2))^(n^2/2) We conjecture that the bound is sharp. Our main tool is the entropy method.Comment: 13 page

New bounds on the number of n-queens configurations

In how many ways can $n$ queens be placed on an $n \times n$ chessboard so that no two queens attack each other? This is the famous $n$-queens problem. Let $Q(n)$ denote the number of such configurations, and let $T(n)$ be the number of configurations on a toroidal chessboard. We show that for every $n$ of the form $4^k+1$, $T(n)$ and $Q(n)$ are both at least $n^{\Omega(n)}$. This result confirms a conjecture of Rivin, Vardi and Zimmerman for these values of $n$. We also present new upper bounds on $T(n)$ and $Q(n)$ using the entropy method, and conjecture that in the case of $T(n)$ the bound is asymptotically tight. Along the way, we prove an upper bound on the number of perfect matchings in regular hypergraphs, which may be of independent interest

Counting Skolem Sequences

We compute the number of solutions to the Skolem pairings problem, S(n), and to the Langford variant of the problem, L(n). These numbers correspond to the sequences A059106, and A014552 in Sloane's Online Encyclopedia of Integer Sequences. The exact value of these numbers were known for any positive integer n < 24 for the first sequence and for any positive integer n < 27 for the second sequence. Our first contribution is computing the exact number of solutions for both sequences for any n < 30. Particularly, we report that S(24) = 102, 388, 058, 845, 620, 672. S(25) = 1, 317, 281, 759, 888, 482, 688. S(28) = 3, 532, 373, 626, 038, 214, 732, 032. S(29) = 52, 717, 585, 747, 603, 598, 276, 736. L(27) = 111, 683, 611, 098, 764, 903, 232. L(28) = 1, 607, 383, 260, 609, 382, 393, 152. Next we present a parallel tempering algorithm for approximately counting the number of pairings. We show that the error is less than one percent for known exact numbers, and obtain approximate values for S(32) ~ 2.2x10^26 , S(33) ~ 3.6x10^27, L(31) ~ 5.3x10^24, and L(32) ~ 8.8x10^2

On the number of SQSs, latin hypercubes and MDS codes

It is established that the logarithm of the number of latin $d$-cubes of order $n$ is $\Theta(n^{d}\ln n)$ and the logarithm of the number of pairs of orthogonal latin squares of order $n$ is $\Theta(n^2\ln n)$. Similar estimations are obtained for systems of mutually strong orthogonal latin $d$-cubes. As a consequence, it is constructed a set of Steiner quadruple systems of order $n$ such that the logarithm of its cardinality is $\Theta(n^3\ln n)$ as $n\rightarrow\infty$ and $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.Comment: 10 page

On the numbers of 1-factors and 1-factorizations of hypergraphs

A 1-factor of a hypergraph $G=(X,W)$ is a set of hyperedges such that every vertex of $G$ is incident to exactly one hyperedge from the set. A 1-factorization is a partition of all hyperedges of $G$ into disjoint 1-factors. The adjacency matrix of a $d$-uniform hypergraph $G$ is the $d$-dimensional (0,1)-matrix of order $|X|$ such that an element $a_{\alpha_1, \ldots, \alpha_d}$ of $A$ equals 1 if and only if $\left\{\alpha_1, \ldots, \alpha_d\right\}$ is a hyperedge of $G$. Here we estimate the number of 1-factors of uniform hypergraphs and the number of 1-factorizations of complete uniform hypergraphs by means of permanents of their adjacency matrices

Efficient Generation of One-Factorizations through Hill Climbing

It is well known that for every even integer $n$, the complete graph $K_{n}$ has a one-factorization, namely a proper edge coloring with $n-1$ colors. Unfortunately, not much is known about the possible structure of large one-factorizations. Also, at present we have only woefully few explicit constructions of one-factorizations. Specifically, we know essentially nothing about the {\em typical} properties of one-factorizations for large $n$. Suppose that $\cal C_{\rm n}$ is a graph whose vertex set includes the set of all order-$n$ one-factorizations and that $\Psi: V(\cal C_{\rm n})\to \mathbb R$ takes its minimum precisely at the one-factorizations. Given $\cal C_{\rm n}$ and $\Psi$, we can generate one-factorizations via hill climbing. Namely, by taking a walk on $\cal C_{\rm n}$ that tends to go from a vertex to a neighbor of smaller $\Psi$. For over 30 years, hill-climbing has been essentially the only method for generating many large one-factorizations. However, the validity of such methods was supported so far only by numerical evidence. Here, we present for the first time hill-climbing algorithms that provably generate an order-$n$ one-factorization in $\text{polynomial}(n)$ steps regardless of the starting state, while all vertex degrees in the underlying graph are appropriately bounded. We also raise many questions and conjectures regarding hill-climbing methods and concerning the possible and typical structure of one-factorizations

A proof of Tomescu's graph coloring conjecture

In 1971, Tomescu conjectured that every connected graph $G$ on $n$ vertices with chromatic number $k\geq4$ has at most $k!(k-1)^{n-k}$ proper $k$-colorings. Recently, Knox and Mohar proved Tomescu's conjecture for $k=4$ and $k=5$. In this paper, we complete the proof of Tomescu's conjecture for all $k\ge 4$, and show that equality occurs if and only if $G$ is a $k$-clique with trees attached to each vertex.Comment: Adds a short proof of the case k=4, removing dependence on previous wor

1-factorizations of pseudorandom graphs

A $1$-factorization of a graph $G$ is a collection of edge-disjoint perfect matchings whose union is $E(G)$. A trivial necessary condition for $G$ to admit a $1$-factorization is that $|V(G)|$ is even and $G$ is regular; the converse is easily seen to be false. In this paper, we consider the problem of finding $1$-factorizations of regular, pseudorandom graphs. Specifically, we prove that an $(n,d,\lambda)$-graph $G$ (that is, a $d$-regular graph on $n$ vertices whose second largest eigenvalue in absolute value is at most $\lambda$) admits a $1$-factorization provided that $n$ is even, $C_0\leq d\leq n-1$ (where $C_0$ is a universal constant), and $\lambda\leq d^{1-o(1)}$. In particular, since (as is well known) a typical random $d$-regular graph $G_{n,d}$ is such a graph, we obtain the existence of a $1$-factorization in a typical $G_{n,d}$ for all $C_0\leq d\leq n-1$, thereby extending to all possible values of $d$ results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed $d$. Moreover, we also obtain a lower bound for the number of distinct $1$-factorizations of such graphs $G$ which is off by a factor of $2$ in the base of the exponent from the known upper bound. This lower bound is better by a factor of $2^{nd/2}$ than the previously best known lower bounds, even in the simplest case where $G$ is the complete graph. Our proofs are probabilistic and can be easily turned into polynomial time (randomized) algorithms

On a conjecture of Erd\H{o}s on locally sparse Steiner triple systems

A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erd\H{o}s conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the aim is to have at most $j-3$ triples on every set of $j$ points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and R\"odl as well as Ellis and Linial in a strong form, and answers a question of Krivelevich, Kwan, Loh, and Sudakov. Moreover, we pose a conjecture which would generalize the Erd\H{o}s conjecture to Steiner systems with arbitrary parameters and provide some evidence for this.Comment: updated references, to appear in Combinatoric

Smoothed counting of 0-1 points in polyhedra

Given a system of linear equations $\ell_i(x)=\beta_i$ in an $n$-vector $x$ of 0-1 variables, we compute the expectation of $\exp\left\{- \sum_i \gamma_i \left(\ell_i(x) - \beta_i\right)^2\right\}$, where $x$ is a vector of independent Bernoulli random variables and $\gamma_i >0$ are constants. The algorithm runs in quasi-polynomial $n^{O(\ln n)}$ time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising model with a sufficiently strong external field. We discuss applications to (perfect) matchings in hypergraphs and randomized rounding in discrete optimization.Comment: 24 pages, several improvement