We show that, for each fixed k, an n-vertex graph not containing a cycle of length 2k has at most $80\sqrt{k\log k}\cdot n^{1+1/k}+O(n)$ edges.

The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied by a random fitness. We concentrate on the typical behaviour of the graph by calculating the fitness distribution of a vertex chosen proportional to its degree. For a particular variant of the model, this analysis was first carried out by Borgs, Chayes, Daskalakis and Roch. However, we present a new method, which is robust in the sense that it does not depend on the exact specification of the attachment law. In particular, we show that a peculiar phenomenon, referred to as Bose–Einstein condensation, can be observed in a wide variety of models. Finally, we also compute the joint degree and fitness distribution of a uniformly chosen vertex.

Finding a hidden partition in a random environment is a general and important problem which contains as subproblems many important questions, such as finding a hidden clique, finding a hidden colouring, finding a hidden bipartition, etc.

In this paper we provide a simple SVD algorithm for this purpose, addressing a question of McSherry. This algorithm is easy to implement and works for sparse graphs under optimal density assumptions. We also consider an approximating algorithm, which on one hand works under very mild assumptions, but on other hand can sometimes be upgraded to give the exact solution.

We show that the number of unit distances determined by n points in ℝ3 is O(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].

Any function F: {0,. . ., N − 1} → {−1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Möbius function μ in the sense that

\[ \frac{1}{N} \sum_{0 \leq x \leq N-1} \mu(x)F(x) → 0 \quad\text{as}~~ N → \infty. \]

Let be the hypergraph whose points are the subsets X of [n] := {1,…,n} with l≤ |X| ≤ u, l < u, and whose edges are intervals in the Boolean lattice of the form I = {C ⊆[n] : X⊆C⊆Y} where |X| = l, |Y| = u, X ⊆ Y.We study the matching number i.e. the the maximum number of pairwise disjoint edges, and the covering number i.e. the minimum number of points which cover all edges. We prove that max and that for every ε > 0 the inequalities hold, where for the lower bounds we suppose that n is not too small. The corresponding fractional numbers can be determined exactly. Moreover, we show by construction that

In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r ≥ 2, there exist constants c, C > 0 such that

$$ \begin{equation*} \Pr[G_{n,p} \rightarrow (F)_r^e] = \begin{cases} 1-o(1) &p\ge Cn^{-1/m_2(F)},\\ o(1) &p\le cn^{-1/m_2(F)}, \end{cases} \end{equation*} $$

$$ \begin{equation*} m_2(F) = \max_{J\subseteq F, v_J\ge 2} \frac{e_J-1}{v_J-2}. \end{equation*} $$

We study the concept of list total colourings and prove that every multigraph of maximum degree 3 is 5-total-choosable. We also show that the total choice number of a graph of maximum degree 2 is equal to its total chromatic number.

A graph is vertex-transitive (edge-transitive) if its automorphism group acts transitively on the vertices (edges, resp.). The expansion rate of a subset S of the vertex set is the quotient e(S):= |∂(S)|/|S|, where ∂(S) denotes the set of vertices not in S but adjacent to some vertex in S. Improving and extending previous results of Aldous and Babai, we give very simple proofs of the following results. Let X be a (finite or infinite) vertex-transitive graph and let S be a finite subset of the vertices. If X is finite, we also assume |S| ≤|V(X)/2. Let d be the diameter of S in the metric induced by X. Then e(S) ≥1/(d + 1); and e(S) ≥ 2/(d +2) if X is finite and d is less than the diameter of X. If X is edge-transitive then |δ(S)|/|S| ≥ r/(2d), where ∂(S) denotes the set of edges joining S to its complement and r is the harmonic mean of the minimum and maximum degrees of X. – Diverse applications of the results are mentioned.

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a class of labelled graphs has smooth growth; and, for the random graph Rn sampled uniformly from the n-vertex graphs in , the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, as n → ∞ the probability that Rn is connected tends to 1/A(ρ), where A(x) is the exponential generating function for and ρ is its radius of convergence.

For an increasing monotone graph property the local resilience of a graph G with respect to is the minimal r for which there exists a subgraph H ⊆ G with all degrees at most r, such that the removal of the edges of H from G creates a graph that does not possess . This notion, which was implicitly studied for some ad hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model (n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random regular graphs of constant degree. We investigate the local resilience of the typical random d-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values of d, with high probability, the local resilience of the random d-regular graph, n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model (n, p), for every positive ϵ > 0 and large enough values of K, if p > then, with high probability, the local resilience of (n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that if d is large enough then, with high probability, a typical random d-regular graph G is such that, in the unbiased Maker–Breaker game played on the edges of G, Maker has a winning strategy to create a Hamilton cycle.

This paper has two main focal points. We first consider an important class of machine learning algorithms: large margin classifiers, such as Support Vector Machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas.

In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether PSPACE can be separated from the polynomial hierarchy in communication complexity.

Communication is a key ingredient in many types of learning. This explains the relations between the field of learning theory and that of communication complexity [6, l0, 16, 26]. The results of this paper constitute another link in this rich web of relations. These new results have already been applied toward the solution of several open problems in communication complexity [18, 20, 29].

We continue the study of random lifts of graphs initiated in [4]. Here we study the possibility of generating graphs with high edge expansion as random lifts. Along the way, we introduce the method of $\epsilon$-nets into the study of random structures. This enables us to improve (slightly) the known bounds for the edge expansion of regular graphs.

We introduce a polynomial invariant of graphs on surfaces, PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for PG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, PG specializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomial PG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.

For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows.

In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree.

In the second part of this paper, for the purpose of computation, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph. We include some applications to graph theory.

We study the following one-person game against a random graph process: the Player's goal is to 2-colour a random sequence of edges of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.

While it is not hard to prove that the expected length of this game is about , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with edges, where c is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round.

Let $F(\b{z})=\sum_\b{r} a_\b{r}\b{z^r}$ be a multivariate generating function that is meromorphic in some neighbourhood of the origin of $\mathbb{C}^d$, and let $\sing$ be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of $\sing$.

In the first article in this series, we treated the case of smooth points of $\sing$. In this article we deal with multiple points of $\sing$. Our results show that the central limit (Ornstein–Zernike) behaviour typical of the smooth case does not hold in the multiple point case. For example, when $\sing$ has a multiple point singularity at $(1, \ldots, 1)$, rather than $a_\b{r}$ decaying as $|\b{r}|^{-1/2}$ as $|\b{r}| \to \infty$, $a_\b{r}$ is very nearly polynomial in a cone of directions.

We consider ‘unconstrained’ random k-XORSAT, which is a uniformly random system of m linear non-homogeneous equations in $\mathbb{F}$2 over n variables, each equation containing k ⩾ 3 variables, and also consider a ‘constrained’ model where every variable appears in at least two equations. Dubois and Mandler proved that m/n = 1 is a sharp threshold for satisfiability of constrained 3-XORSAT, and analysed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT.

We show that m/n = 1 remains a sharp threshold for satisfiability of constrained k-XORSAT for every k ⩾ 3, and we use standard results on the 2-core of a random k-uniform hypergraph to extend this result to find the threshold for unconstrained k-XORSAT. For constrained k-XORSAT we narrow the phase transition window, showing that m − n → −∞ implies almost-sure satisfiability, while m − n → +∞ implies almost-sure unsatisfiability.

We derive improved isoperimetric inequalities for discrete product measures on the n-dimensional cube. As a consequence, a general theorem on the threshold behaviour of monotone properties is obtained. This is then applied to coding theory when we study the probability of error after decoding.