We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth $3$; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$); and the class of $K_t$-minor-free graphs has underlying treewidth $t-2$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

A relationship between a new and an old graph invariant is established. The first invariant is connected to the ‘sandglass conjecture’ of [1]. The second one is graph entropy, an information theoretic functional, which is already known to be relevant in several combinatorial contexts.

We give an asymptotic formula for the minimum number of edges contained in triangles among graphs with n vertices and e edges. Our main tool is a generalization of Zykov's symmetrization method that can be applied to several graphs simultaneously.

We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two points x and y are connected with probability g(y−x), where g is a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.

We show that if S1 is a strongly complete sum-free set of positive integers, and if S0 is a finite sum-free set, then, with positive probability, a random sum-free set U contains S0 and is contained in S0∪S1. As a corollary we show that, with positive probability, 2 is the only even element of a random sum-free set.

Given a digraph D, let δ0(D) := min{δ+(D), δ−(D)} be the minimum semi-degree of D. We show that every sufficiently large digraph D with δ0(D)≥n/2 + l −1 is l-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis [17]. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e., that for every sequence s1, . . ., sk of distinct vertices of D there is a directed cycle which encounters s1, . . ., sk in this order. This result will be used in [16].

Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of weighted combinatorial objects. We describe a general setting characterized by the formation of a giant component. The collection of small fragments is shown to converge in total variation toward a limit object following a Pólya–Boltzmann distribution.

The maximum size of an r-uniform hypergraph without a Berge cycle of length at least k has been determined for all k ≥ r + 3 by Füredi, Kostochka and Luo and for k < r (and k = r, asymptotically) by Kostochka and Luo. In this paper we settle the remaining cases: k = r + 1 and k = r + 2, proving a conjecture of Füredi, Kostochka and Luo.

An [n, k, r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, V1, …, Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 [les ] i < j [les ] n. An independent transversal in an [n, k, r]-partite graph is an independent set, T, consisting of n vertices, one from each Vi. An independent covering is a set of k pairwise-disjoint independent transversals. Let t(k, r) denote the maximal n for which every [n, k, r]-partite graph contains an independent transversal. Let c(k, r) be the maximal n for which every [n, k, r]-partite graph contains an independent covering. We give upper and lower bounds for these parameters. Furthermore, our bounds are constructive. These results improve and generalize previous results of Erdo″s, Gyárfás and Łuczak [5], for the case of graphs.

For every n consider a subset Hn of the patterns of length n over a fixed finite alphabet. The limit distribution of the waiting time until each element of Hn appears in an infinite sequence of independent, uniformly distributed random letters was determined in an earlier paper. This time we prove that these waiting times are getting independent as n → ∞. Our result is used for applying the converse part of the Borel–Cantelli lemma to problems connected with such waiting times, yielding thus improvements on some known theorems.

A graph construction game is a Maker–Breaker game. Maker and Breaker take turns in choosing previously unoccupied edges of the complete graph KN. Maker's aim is to claim a copy of a given target graph G while Breaker's aim is to prevent Maker from doing so. In this paper we show that if G is a d-degenerate graph on n vertices and N > d1122d+9n, then Maker can claim a copy of G in at most d1122d+7n rounds. We also discuss a lower bound on the number of rounds Maker needs to win, and the gap between these bounds.

The partition functions of the Ising and Potts models in statistical mechanics are well known to be partial evaluations of the Tutte–Whitney polynomial of the appropriate graph. The Ashkin–Teller model generalizes the Ising model and the four-state Potts model, and has been extensively studied since its introduction in 1943. However, its partition function (even in the symmetric case) is not a partial evaluation of the Tutte–Whitney polynomial. In this paper, we show that the symmetric Ashkin–Teller partition function can be obtained from a generalized Tutte–Whitney function which is intermediate in a precise sense between the usual Tutte–Whitney polynomialof the graph and that of its dual.

The size-Ramsey number of a graph F is the smallest number of edges in a graph G with the Ramsey property for F, that is, with the property that any 2-colouring of the edges of G contains a monochromatic copy of F. We prove that the size-Ramsey number of the grid graph on n × n vertices is bounded from above by n3+o(1).

Let D1 and D2 be two dice with k and l integer faces, respectively, where k and l are two positive integers. The game Gn consists of tossing each die n times and summing the resulting faces. The die with the higher total wins the game. We examine the question of which die wins game Gn more often, for large values of n. We also give an example of a set of three dice which is non-transitive in game Gn for infinitely many values of n.

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$ -regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $2$ . The second is that every graph on $n$ vertices with minimum degree $\delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{\delta +1}+2$ . Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices $n$ . In particular we show that if $d^3 \log n \leq o(n)$ then every $d$ -regular graph with $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ is $(1+o(1))\frac{n}{d+1}$ . We also prove that any graph with $n$ vertices and minimum degree $\delta$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))\frac{n}{\delta +1}+2$ times.

A function f: 2n → {0,1} is odd-cycle-free if there are no x1,. . .,xk ∈ 2n with k an odd integer such that f(x1) = ··· = f(xk) = 1 and x1 + ··· + xk = 0. We show that one can distinguish odd-cycle-free functions from those ε-far from being odd-cycle-free by making poly(1/ε) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs.

The first issue we study is directly reducing testing of linear-invariant properties of Boolean functions to testing associated graph properties. We show a black-box reduction from testing odd-cycle-freeness to testing bipartiteness of graphs. Such reductions have already been shown (Král’, Serra and Vena, and Shapira) for monotone linear-invariant properties defined by forbidding solutions to a finite number of equations. But for odd-cycle-freeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linear-invariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the one-sided testable linear-invariant properties, an open problem raised by Sudan.

The second issue we study is whether there is an efficient canonical tester for linear-invariant properties of Boolean functions. A canonical tester for linear-invariant properties operates by picking a random linear subspace and then checking whether the restriction of the input function to the subspace satisfies a fixed property. The question is if, for every linear-invariant property, there is a canonical tester for which there is only a polynomial blow-up from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question remains open.

We prove an essentially sharp $\tilde \Omega (n/k)$ lower bound on the k-round distributional complexity of the k-step pointer chasing problem under the uniform distribution, when Bob speaks first. This is an improvement over Nisan and Wigderson’s $\tilde \Omega (n/{k^2})$ lower bound, and essentially matches the randomized lower bound proved by Klauck. The proof is information-theoretic, and a key part of it is using asymmetric triangular discrimination instead of total variation distance; this idea may be useful elsewhere.

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the largest component in such variations of the Erdős–Rényi random graph process has recently received considerable attention, in particular for Bollobás's ‘product rule’. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the ‘giant’ component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes.

Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.

We show that the complete list of regular excluded minors for the class of signed-graphic matroids is M*(G1),. . . , M*(G29), R15, R16. Here G1,. . . , G29 are the vertically 2-connected excluded minors for the class of projective-planar graphs and R15 and R16 are two regular matroids that we will define in the article.