Let G be a finite graph with minimum degree r. Form a random subgraph Gp of G by taking each edge of G into Gp independently and with probability p. We prove that for any constant ε > 0, if $p=\frac{1+\epsilon}{r}$, then Gp is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs.

In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube, where there is a clear definition of influence, in the general case several definitions have been presented in different papers. We propose a family of definitions for the influence that contains all the known definitions, as well as other natural definitions, as special cases. We show that the proofs of the BKKKL theorem and of other results can be adapted to our new definition. The adaptation leads to generalizations of these theorems, which are tight in terms of the definition of influence used in the assertion.

Let graph $G=(V,E)$ and integer $b\geq 1$ be given. A set $S\subseteq V$ is said to be $b$-independent if $u,v\in S$ implies $d_G(u,v)>b$, where $d_G(u,v)$ is the shortest distance between $u$ and $v$ in $G$. The $b$-independence number $\a_b(G)$ is the size of the largest $b$-independent subset of $G$. When $b=1$ this reduces to the standard definition of independence number.

We study this parameter in relation to the random graph $G_{n,p},\,p=d/n$, in particular, when $d$ is a large constant. We show that w.h.p. if $d\geq d_{\epsilon, b}$, $$ \left| \alpha_b(G_{n,p}) - \frac{2bn}{d^b} \biggl(\log{d} - \frac{\log{\log{d}}}{b} - \frac{\log{2b}}{b} + \frac{1}{b}\biggr)\right| \leq \frac{\epsilon n}{d^b}.$$

Let be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality || ≤ 2n−1. Suppose that ||=2n−1 + i. Choose the members of independently with probability p (delete them with probability 1 − p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all -element subsets. We determine the most probably inclusion-free family too, when the number of members is .

We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n → ∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial.

Let m(k) denote the maximum number of edges in a non-extendable, intersecting k-graph. Erdős and Lovász proved that m(k) ≤ kk. For k ≥ 625 we prove m(k) < kk・e−k1/4/6.

We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

It is known that random $k$-Sat instances with at least $(2^k \cdot \ln 2)\cdot n$ random clauses are unsatisfiable with high probability. This result is simply obtained by showing that the expected number of satisfying assignments tends to $0$ when the number of variables $n$ tends to infinity. This proof does not directly provide us with an efficient algorithm certifying the unsatisfiability of a given random formula. Concerning efficient algorithms, it is essentially known that random formulas with $n^\varepsilon \cdot n^{k/2}$ clauses with $k$ literals can be efficiently certified as unsatisfiable. The present paper is the result of trying to lower this bound. We obtain better bounds for some specialized satisfiability problems. These results are based on discrepancy investigations for hypergraphs.

Further, we show that random formulas with a linear number of clauses can be efficiently certified as unsatisfiable in the Not-All-Equal-$3$-Sat sense. A similar result holds for the non-$3$-colourability of random graphs with a linear number of edges. We obtain these results by direct application of approximation algorithms.

A graph Γ with diameter d is strongly distance-regular if Γ is distance-regular and its distance-d graph Γd is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph Γd = Γ (the complement of Γ), and the antipodal distance-regular graphs. Here we study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph Γ, with distinct eigenvalues λ0 > λ1 > λ2 > λ3, is strongly distance-regular if and only if λ2 = −1, and Γ3 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of Γ.

We consider the problem of fault diagnosis in multiprocessor systems. Processors perform tests on one another: fault-free testers correctly identify the fault status of tested processors, while faulty testers can give arbitrary test results. Processors fail independently with constant probability p<1/2 and the goal is to identify correctly the status of all processors, based on the set of test results. For 0<q<1, q-diagnosis is a fault diagnosis algorithm whose probability of error does not exceed q. We show that the minimum number of tests to perform q-diagnosis for n processors is Θ(n log 1/q) in the nonadaptive case and n+Θ( log 1/q) in the adaptive case. We also investigate q-diagnosis algorithms that minimize the maximum number of tests performed by, and performed on, processors in the system, constructing testing schemes in which each processor is involved in very few tests. Our results demonstrate that the flexibility yielded by adaptive testing permits a significant saving in the number of tests for the same reliability of diagnosis.

A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$. The previous best bound was $O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$. The best previous bound for this result was exponential in $\Delta$.

A celebrated theorem of Turán asserts that every graph on n vertices with more than $\frac{r\,{-}\,1}{2r}n^2$ edges contains a copy of a complete graph $K_r+1$. In this paper we consider the following more general question. Let G be a $K_r+1-free graph of order n and let α be a constant, 0<α≤1. How dense can every induced subgraph of G on αn vertices be? We prove the following local density extension of Turán's theorem.

For every integer $r\geq 2$ there exists a constant $c_r < 1$ such that, if $c_r < \alpha < 1$ and every αn vertices of G span more than

\[ \frac{r\,{-}\,1}{2r}(2\alpha -1)n^2\vspace*{7pt} \]

edges, then G contains a copy of $K_r+1$. This result is clearly best possible and answers a question of Erdős, Faudree, Rousseau and Schelp [5].

In addition, we prove that the only $K_r+1-free graph of order n, in which every αn vertices span at least $\frac{r\,{-}\,1}{2r}(2\alpha -1)n^2$ edges, is a Turán graph. We also obtain the local density version of the Erdős–Stone theorem.

We consider a problem introduced by Mossel and Ross (‘Shotgun assembly of labeled graphs’, arXiv:1504.07682). Suppose a random n × n jigsaw puzzle is constructed by independently and uniformly choosing the shape of each ‘jig’ from q possibilities. We are given the shuffled pieces. Then, depending on q, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of a puzzle are similar if they only differ by a global rotation of the puzzle, permutation of duplicate pieces, and rotation of rotationally symmetric pieces. In this paper, we show that, with high probability, such a puzzle has at least two non-similar solutions when 2 ⩽ q ⩽ 2e−1/2n, all solutions are similar when q ⩾ (2+ϵ)n, and the solution is unique when q = ω(n).

We show that, if M is a connected binary matroid of cogirth at least five which does not have both an F7-minor and an F*7-minor, then M has a circuit C such that M − C is connected and r(M − C) = r(M).

In this paper we prove that, if a collection of sets [Ascr ] is union-closed, then the average set size of $\cal A is at least $\frac{1}{2}\log_2 (\vert{\cal A}\vert)$.

For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ ℕ to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by Fuchs, we obtain results for the range of small k-values and the range of k-values proportional to the size n of T. In both cases emphasis is on the process view, i.e., the joint distributions for several k-values. We also show that the dynamics of the tree sequence lead to a qualitative difference between the asymptotic behaviour of the lower and the upper end of the profile.

Consider two graphs G1 and G2 on the same vertex set V and suppose that Gi has mi edges. Then there is a bipartition of V into two classes A and B so that, for both i = 1, 2, we have . This gives an approximate answer to a question of Bollobás and Scott. We also prove results about partitions into more than two vertex classes. Our proofs yield polynomial algorithms.

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.

In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we call cores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. If Bn is a graph drawn uniformly at random from a suitable class of labelled biconnected graphs, then we show that with probability 1 − o(1) as n → ∞, Bn belongs to exactly one of the following categories:

1. (i) either there is a unique giant core in Bn, that is, there is a 0 < c = c() < 1 such that the largest core contains ~ cn vertices, and every other core contains at most nα vertices, where 0 < α = α() < 1;

2. (ii) or all cores of Bn contain O(logn) vertices.

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

We show that, for any finite set P of points in the plane and for any integer k ≥ 2, there is a finite set R = R(P, k) with the following property: for any k-colouring of R there is a monochromatic set , ⊆ R, such that is combinatorially equivalent to the set P, and the convex hull of P contains no point of R \ . We also consider related questions for colourings of p-element subsets of R (p > 1), and show that these analogues have negative solutions.