It is proved that the median eigenvalues of every connected bipartite graph G of maximum degree at most three belong to the interval [−1, 1] with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$. Moreover, if G is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval [−1, 1]. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.

The main result of this paper gives a structure of a triangle-free graph of order n with minimal degree greater than 10n/29. This extends results given by Andrásfai et al. [1] and Häggkvist [6].

Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).

To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).

We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

In a random passive intersection graph model the edges of the graph are decided by taking the union of a fixed number of cliques of random size. We give conditions for a random passive intersection graph model to have a limiting vertex degree distribution, in particular to have a Poisson limiting vertex degree distribution. We give related conditions which, in addition to implying a limiting vertex degree distribution, imply convergence of expectation.

The structure of non-Hamiltonian graphs is studied in terms of neighbourhood-unions. Numerous results, such as (A), (B) and (C) below, are obtained.

In this paper we prove that two local conditions involving the degrees and co-degrees in a graph can be used to determine whether a given vertex partition is Frieze–Kannan regular. With a more refined version of these two local conditions we provide a deterministic algorithm that obtains a Frieze–Kannan regular partition of any graph G in time O(|V(G)|2).

Mader asked whether every $C_4$-free graph $G$ contains a subdivision of a complete graph whose order is at least linear in the average degree of $G$. We show that there is a subdivision of a complete graph whose order is almost linear. More generally, we prove that every $K_{s,t}$-free graph of average degree $r$ contains a subdivision of a complete graph of order $r^{\frac{1}{2}{+}\frac{1}{2(s-1)}-o(1)}$.

Let S be a set of n points in ${\mathbb R}^{2}$ contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least cdn4/3, unless C contains a line or a circle.

We also prove the lower bound cd′ min{m2/3n2/3, m2, n2} for the number of distinct distances between m points on one irreducible plane algebraic curve and n points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let k ≥ 2 be an integer and let G be a hypergraph with mi edges of size i for i=1,2. Bollobás and Scott conjectured that G has a partition into k classes, each of which contains at most $m_1/k+m_2/k^2+O(\sqrt{m_1+m_2})$ edges. In this paper, we confirm the conjecture affirmatively by showing that G has a partition into k classes, each of which contains at most

$$m_1/k+m_2/k^2+\ffrac{k-1}{2k^2}\sqrt{2(km_1+m_2)}+O(1)$$.

We use a greedy probabilistic method to prove that, for every ε > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1 − ε)n. That is, we show the existence of a second Latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.

Let C(G) denote the number of simple cycles of a graph G and let C(n) be the maximum of C(G) over all planar graphs with n nodes. We present a lower bound on C(n), constructing graphs with at least 2.28n cycles. Applying some probabilistic arguments we prove an upper bound of 3.37n.

We also discuss this question restricted to the subclasses of grid graphs, bipartite graphs, and 3-colourable triangulated graphs.

We prove tail estimates for variables of the form ∑if(Xi), where (Xi)i is a sequence of states drawn from a reversible Markov chain, or, equivalently, from a random walk on an undirected graph. The estimates are in terms of the range of the function f, its variance, and the spectrum of the graph. The purpose of our estimates is to determine the number of chain/walk samples which are required for approximating the expectation of a distribution on vertices of a graph, especially an expander. The estimates must therefore provide information for fixed number of samples (as in Gillman's [4]) rather than just asymptotic information. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein- and Bennett-type inequalities, as well as an inequality for sub-Gaussian variables.

This paper gives a general expression for the Tutte polynomial of perfect matroid designs, that is, for matroids in which all ats of the same rank are equicardinal. In particular, explicit expressions are given for the Tutte polynomial of projective and affine geometries over a finite field.

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős–Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ)-distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomized algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.

Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark examples for algorithms, and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution (Krzakala et al. 2007, Proc. Nat. Acad. Sci.), physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications for Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g. Banks et al. 2016, Proc. 29th COLT).

A set of n triangles sharing a common edge is called a book with n pages and is denoted by $B_{n}$. It is known that the Ramsey number $r ( B_{n} ) $ satisfies $r ( B_{n} ) = ( 4+o ( 1 ) ) n.$ We show that every red–blue edge colouring of $K_{ \lfloor ( 4-\varepsilon ) n \rfloor }$ with no monochromatic $B_{n}$ exhibits quasi-random properties when $\varepsilon$ tends to 0. This implies that there is a constant $c>0$ such that for every red–blue edge colouring of $K_{r ( B_{n} ) }$ there is a monochromatic $B_{n}$ whose vertices span at least $ \lfloor cn^{2} \rfloor $ edges of the same colour as the book.

As an application we find the Ramsey number for a class of graphs.

Häggkvist and Scott asked whether one can find a quadratic function q(k) such that, if G is a graph of minimum degree at least q(k), then G contains vertex-disjoint cycles of k consecutive even lengths. In this paper, it is shown that if G is a graph of average degree at least k2+19k+10 with sufficiently many vertices, then G contains vertex-disjoint cycles of k consecutive even lengths, answering the above question in the affirmative. The coefficient of k2 cannot be decreased and, in this sense, this result is best possible.

Let G denote a finite abelian group of order n and Davenport constant D, and put m = n + D − 1. Let x = (x1,. . .,xm) ∈ Gm. Gao's theorem states that there is a reordering (xj1, . . ., xjm) of x such that

Let w = (x1, . . ., wm) ∈ ℤm. As a corollary of the main result, we show that there are reorderings (xj1, . . ., xjm) of x and (wk1, . . ., wkm) of w, such that where xj1 is the most repeated value in x. For w = (1, . . ., 1), this result reduces to Gao's theorem.

In this paper we study a question related to the classical Erdős–Ko–Rado theorem, which states that any family of k-element subsets of the set [n] = {1,. . .,n} in which any two sets intersect has cardinality at most $\binom{n-1}{k-1}$.

We say that two non-empty families ${\mathcal A}, {\mathcal B}\subset \binom{[n]}{k}$ are s-cross-intersecting if, for any A ∈ ${\mathcal A}$, B ∈ ${\mathcal B}$, we have |A ∩ B| ≥ s. In this paper we determine the maximum of |${\mathcal A}$|+|${\mathcal B}$| for all n. This generalizes a result of Hilton and Milner, who determined the maximum of |${\mathcal A}$|+|${\mathcal B}$| for non-empty 1-cross-intersecting families.