Abstract: Let r = r(n) → ∞ with 3 l r l n1−η for an arbitrarily small constant η > 0, and let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set l1, 2, …, nr. We prove that, with probability tending to 1 as n → ∞, Gr has the following properties: the independence number of Gr is asymptotically 2n log r/r and the chromatic number of Gr is asymptotically r/2nlogr.

Abstract: We study the complexity of computing the coefficients of three classical polynomials, namely the chromatic, flow and reliability polynomials of a graph. Each of these is a specialization of the Tutte polynomial Σtijxiyj. It is shown that, unless NP = RP, many of the relevant coefficients do not even have good randomized approximation schemes. We consider the quasi-order induced by approximation reducibility and highlight the pivotal position of the coefficient t10 = t01, otherwise known as the beta invariant.Our nonapproximability results are obtained by showing that various decision problems based on the coefficients are NP-hard. A study of such predicates shows a significant difference between the case of graphs, where, by Robertson–Seymour theory, they are computable in polynomial time, and the case of matrices over finite fields, where they are shown to be NP-hard.

Abstract: Given a family F of r-graphs, let ex(n, F) be the maximum number of edges in an n-vertex r-graph containing no member of F. Let C(r) 4 denote the family of r-graphs with distinct edges A, B, C, D, such that A ∩ B = C ∩ D = O and A ∪ B = C ∪ D. For s1 l … l sr, let K(r) (s1,…,sr) be the complete r-partite r-graph with parts of sizes s1,…,sr.Furedi conjectured over 15 years ago that ex(n,C(3)4) l (n2) for sufficiently large n. We prove the weaker resultex(n, {C(3)4, K(3)(1,2,4)}) l (n2).Generalizing a well-known conjecture for the Turan number of bipartite graphs, we conjecture thatex(n, K(r)(s1,…,sr)) = Θ(nr−1/s),where s = Πr−1i=1 si. We prove this conjecture when s1 = … = sr−2 = 1 and(i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii)sr > (sr−1−1)!.In cases (i) and (ii), we determine the asymptotic value of ex(n,K(r)(s1,…,sr)).We also provide an explicit construction givingex(n,K(3)(2,2,3)) > (1/6−o(1))n8/3.This improves upon the previous best lower bound of O(n29/11) obtained by probabilistic methods. Several related open problems are also presented.

Abstract: We consider a stochastic process based on the iterated prisoner's dilemma game. During the game, each of n players has a state, either cooperate or defect. The players are connected by an ‘interaction graph’. During each step of the process, an edge of the graph is chosen uniformly at random and the states of the players connected by the edge are modified according to the Pavlov strategy. The process converges to a unique absorbing state in which all players cooperate. We prove two conjectures of Kittock: the convergence rate is exponential in n when the interaction graph is a complete graph, and it is polynomial in n when the interaction graph is a cycle. In fact, we show that the rate is O(n log n) in the latter case.

Abstract: For a stochastic approximation-type recursion with finitely many possible limit points, we find a lower bound on the probability of converging to a prescribed point in its ‘domain of attraction’. This has implications for the lock-in phenomena in the stochastic models of increasing return economics and the sample complexity of stochastic approximation algorithms in engineering.

Abstract: In this paper we are concerned with the following conjecture.Conjecture. For any positive integers n and k satisfying k < n, and any sequence a1, a2, … ak of not necessarily distinct elements of Zn, there exists a permutation π ∈ Sk such that the elements aπ(i)+i are all distinct modulo n.We prove this conjecture when 2k ≤ n+1. We then apply this result to tree embeddings. Specifically, we show that, if T is a tree with n edges and radius r, then T decomposes Kt for some t ≤ 32(2r+4)n2+1.

Abstract: We study the asymptotic behaviour of the relative entropy (to stationarity) for a commonly used model for riffle shuffling a deck of n cards m times. Our results establish and were motivated by a prediction in a recent numerical study of Trefethen and Trefethen. Loosely speaking, the relative entropy decays approximately linearly (in m) for m log2 n. The deck becomes random in this information-theoretic sense after m = 3/2 log2 n shuffles.

Abstract: Given a graph G on n vertices with average degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that, if the edge connectivity k(G) satisfies k(G) > d/log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

Abstract: Let c l 0.076122 and T1, T2,…, Tn be a sequence of trees such that mV(Ti)m l i−c(i−1). We prove that, if for each 1 l i l n there exists a vertex xi ∈ V(Ti) such that Ti−xi has at least (1−2c)(i−1) isolated vertices, then T1,…, Tn can be packed into Kn. We also prove that if T is a tree of order n+1−c′n, c′ l 1/25 (37−8 √21 ) a 0.0135748, such that there exists a vertex x ∈ V(T) and T−x has at least n(1−2c′) isolated vertices, then 2np1 copies of T may be packed into K2np1.

Abstract: We present improved lower and upper bounds for the time constant of first-passage percolation on the square lattice. For the case of lower bounds, a new method, using the idea of a transition matrix, has been used. Numerical results for the exponential and uniform distributions are presented. A simulation study is included, which results in new estimates and improved upper confidence limits for the time constants.

Abstract: We show that recognizing the K3-freeness and K4-freeness of graphs is hard, respectively, for two-player nondeterministic communication protocols using exponentially many partitions and for nondeterministic syntactic read-r times branching programs.The key ingredient is a generalization of a colouring lemma, due to Papadimitriou and Sipser, which says that for every balanced redblue colouring of the edges of the complete n-vertex graph there is a set of en2 triangles, none of which is monochromatic, such that no triangle can be formed by picking edges from different triangles. We extend this lemma to exponentially many colourings and to partial colourings.

Abstract: Consider a finite alphabet Ω and patterns which consist of characters from Ω. For a given pattern w, let cor(w) denote its autocorrelation, which can be seen as a measure of the amount of overlap in w. Letting aw(n) denote the number of strings over Ω of length n which do not contain w as a substring, the main result of this paper is: If cor(w) > cor(w′) then aw(n)−aw′(n) > (|Ω|−1)(aw(n−1)−aw′(n−1)) for n ≥ N, and the value of N is given. This result confirms a conjecture by Eriksson [2], which was previously proved to be true by Cakir, Chryssaphinou and Mansson [1] when |Ω| ≥ 3.

Abstract: We consider k-uniform set systems over a universe of size n such that the size of each pairwise intersection of sets lies in one of s residue classes mod q, but k does not lie in any of these s classes. A celebrated theorem of Frankl and Wilson [8] states that any such set system has size at most (ns) when q is prime. In a remarkable recent paper, Grolmusz [9] constructed set systems of superpolynomial size Ω(exp(c log2 n/log log n)) when q = 6. We give a new, simpler construction achieving a slightly improved bound. Our construction combines a technique of Frankl [6] of applying polynomials to set systems with Grolmusz's idea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also extend Frankl's original argument to arbitrary prime-power moduli: for any e > 0, we construct systems of size ns+g(s), where g(s) = Ω(s1−E). Our work overlaps with a very recent technical report by Grolmusz [10].

Abstract: In this paper we study distances in random subgraphs of a generalized n-cube Qns over a finite alphabet S of size s. Qns is the direct product of complete graphs over s vertices, its vertices being the n-tuples (x1, …, xn), with xi ∈ S, i = 1, … n, and two vertices being adjacent if they differ in exactly one coordinate. A random (induced) subgraph γ of Qns is obtained by selecting Qns-vertices with independent probability pn and then inducing the corresponding edges from Qns. Our main result is that dγ (P,Q) l [2k+3]dQns (P,Q) almost surely for P,Q ∈ γ, pn = n−a and 0 l a < ½, where k = [1+3a/1−2a] and dγ and dQns denote the distances in γ and Qns, respectively.

Abstract: Let lAvrv∈V be a finite collection of events and G = (V, E) be a chordal graph. Our main result – the chordal graph sieve – is a Bonferroni-type inequality where the selection of intersections in the estimates is determined by a chordal graph G. It interpolates between Boole's inequality (G empty) and the sieve formula (G complete). By varying G, several inequalities both well-known and new are obtained in a concise and unified way.

Abstract: We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

Abstract: Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = m{S ⊆ Tn : 1n ∈ S, S ≅ T}m, and B(n; T) = m{S ⊆ Tn : 1n ∉ S, S ≅ T}m. In this note we prove that ***** insert equation here ***** for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

Abstract: A top to random shuffle of a deck of cards is performed by taking the top card off of the deck and replacing it in a randomly chosen position of the deck. We find approximations of the relative entropy of a deck of n cards after m successive top to random shuffles. Initially the relative entropy decays linearly and for larger m it decays geometrically at a rate that alters abruptly at m = n log n. It converges to an explicitly given expression when m = [n log n+cn] for a constant c.

Abstract: In this note it is shown that resistance and resistance dimension are rough isometry invariant.

Abstract: We show that, for every positive integer c*, there is an integer n such that, if M is a matroid whose largest cocircuit has size c*, then E(M) can be partitioned into two sets E1 and E2 such that every connected component of each of MmE1 and MmE2 has at most n elements.

Abstract: For two stochastically dependent random variables X and Y taking values in {0,…, m−1}, we study the distribution of the random residue U = XY mod m. Our main result is an upper bound for the distance Δm = supxE[0,1] ȣ P(U/m ≤ x)−xȣ. For independent and uniformly distributed X and Y, the exact distribution of U is derived and shown to be stochastically smaller than the uniform distribution on {0,…, m−1}. Moreover, in this case Δm is given explicitly.

Abstract: The circuit cover problem for mixed graphs (those containing edges and/or arcs) is defined as follows. Given a mixed graph M with a nonnegative integer weight function p on its edges and arcs, decide whether ( M , p ) has a circuit cover , that is, a list of circuits in M such that every edge (arc) e is contained in exactly p ( e ) circuits of the list. In the special case when M is a directed graph (contains only arcs), the problem is easy, but when M is an undirected graph not many results are known. For general mixed graphs this problem was shown to be NP-complete by Arkin and Papadimitriou in 1986. We prove that this problem remains NP-complete for planar mixed graphs. Furthermore, we present a good characterization for the existence of a circuit cover when M is series-parallel (a similar result holds for the fractional version). We also describe a polynomial algorithm to find such a circuit cover, when it exists. This is an ellipsoid-based algorithm whose separation problem is the minimum circuit problem on series-parallel mixed graphs, which we show to be polynomially solvable. Results on two well-known combinatorial problems, the problem of detecting negative circuits and the problem of finding shortest paths, are also presented. We prove that both problems are NP-hard for planar mixed graphs.

Abstract: In ‘Automorphisms of Dowling lattices and related geometries’, J. Bonin constructed the automorphism group A of a Dowling lattice as the image of a certain semidirect product, A = t(K r H). In this work we find necessary and sufficient conditions for this quotient to be the semidirect product A = t(K) r t(H). In addition, we include a construction of A that lends itself to computation more readily than that found in Bonin's work.

Abstract: An instance of the square packing problem consists of n squares with independently, uniformly distributed side-lengths and independently, uniformly distributed locations on the unit d-dimensional torus. A packing is a maximum family of pairwise disjoint squares. The one-dimensional version of the problem is the classical random interval packing problem. This paper deals with the asymptotic behaviour of packings as n tends to infinity while d = 2. Coffman, Lueker, Spencer and Winkler recently proved that the average size of packing is Θ(nd/(d+1)). Using partitioning techniques, sub-additivity and concentration of measure arguments, we show first that, after normalization by n2/3, the size of two-dimensional square packings tends in probability toward a genuine limit γ. Straightforward concentration arguments show that large fluctuations of order n2/3 should have probability vanishing exponentially fast with n2/3. Even though γ remains unknown, using a change of measure argument we show that this upper bound on tail probabilities is qualitatively correct.

Abstract: Consider the class of graphs on n vertices which have maximum degree at most 1/2n−1+τ, where τ ≥ −n1/2+e for sufficiently small e > 0. We find an asymptotic formula for the number of such graphs and show that their number of edges has a normal distribution whose parameters we determine. We also show that expectations of random variables on the degree sequences of such graphs can often be estimated using a model based on truncated binomial distributions.