Abstract: The Turan number of a graph H, ex(n, H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let Pl denote a path on l vertices, and let k ⋠Pl denote k vertex-disjoint copies of Pl. We determine ex(n, k ⋠P3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex(n, k ⋠Pl) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous ErdA‘s-Sos conjecture, and conditional on its truth we determine ex(n, H) when H is an equibipartite forest, for appropriately large n.

Abstract: Let d = (d1, d2,. . ., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph. Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n. Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).

Abstract: Two players share a connected graph with non-negative weights on the vertices. They alternately take the vertices (one in each turn) and collect their weights. The rule they have to obey is that the remaining part of the graph must be connected after each move. We conjecture that the first player can get at least half of the weight of any tree with an even number of vertices. We provide a strategy for the first player to get at least 1/4 of an even tree. Moreover, we confirm the conjecture for subdivided stars. The parity condition is necessary: Alice gets nothing on a three-vertex path with all the weight at the middle. We suspect a kind of general parity phenomenon, namely, that the first player can gather a substantial portion of the weight of any 'simple enough' graph with an even number of vertices.

Abstract: Let (r-1)n â�� rk and let $\FF_1,\ldots,\FF_r\subset\binom{[n]}k$ . Suppose that F1 â�© â��â��â�� â�© Fr â� â�� holds for all Fi â�� i, 1 â�� i â�� r. Then we show that $\prod_{i=1}^r|\FF_i|\leq\binom{n-1}{k-1}^r$ .

Abstract: Let G be a graph with m edges, and let k be a positive integer. We show that V(G) admits a k-partition V1,.i¾ .i¾ . Vk such that $e(V_i)\leq \frac 1{k^2}m+\frac {k-1}{2k^2}(\sqrt{2m+1/4}-1/2)$ for i ∈ {1, 2,.i¾ .i¾ . k}, and $e(V_1, \ldots, V_k)\geq \frac{k-1}{ k} m +\frac{k-1}{ 2k}\sqrt{2m+1/4} +O(k)$ , where e(Vi) denotes the number of edges with both ends in Vi and $e(V_1,\ldots, V_k)=m-\sum_{i=1}^ke(V_i)$ . This answers a problem of Bollobas and Scott [2] in the affirmative. Moreover, $\binom{k+1}{ 2}e(V_i)+\frac k2\sum_{j e i}e(V_j)\le m + O(k^2)$ for i ∈ {1, 2,.i¾ .i¾ ., k}, which is close to being best possible and settles another problem of Bollobas and Scott [2].

Abstract: The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16, 17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to $\frac{5n}{4}$ . The result is proved with the Regularity Lemma via the existence of a monochromatic connected matching covering asymptotically 4n/5 vertices in the multicoloured 2-shadow graph induced by the colouring of Kn(3).

Abstract: Our aim in this note is to make some conjectures about extremal densities of daisy-free families, where a 'daisy' is a certain hypergraph. These questions turn out to be related to some Turan problems in the hypercube, but they are also natural in their own right. We start by giving the daisy conjectures, and some related problems, and shall then go on to describe the connection with vertex-Turan problems in the hypercube.

Abstract: We study a problem on edge percolation on product graphs G × K2. Here G is any finite graph and K2 consists of two vertices {0, 1} connected by an edge. Every edge in G × K2 is present with probability p independent of other edges. The bunkbed conjecture states that for all G and p, the probability that (u, 0) is in the same component as (v, 0) is greater than or equal to the probability that (u, 0) is in the same component as (v, 1) for every pair of vertices u, v ∈ G. We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs G, in particular outerplanar graphs.

Abstract: We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or -1 and if ⊂ ℤ/pℤ is not too large (with respect to p), then | + t ⋠| is significantly larger than 2| | (unless |t| = 3). In the important case |t| = 2, we obtain for instance | + t ⋠| ≥ 2.08 | |-2.

Abstract: 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 B n 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 → ∞, B n belongs to exactly one of the following categories: (i) either there is a unique giant core in B n, that is, there is a 0 ) ) < 1; (ii) or all cores of B n contain O(logn) vertices. Moreover, we find the critical condition that determines the category to which B n belongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 and n. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particular c = 0.765. . . and α = 2/3.

Abstract: We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C4. We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O((nlogn)1/3) and consequently size O(n4/3(logn)1/3), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollobas and Riordan and Osthus and Taraz.

Abstract: Dubhashi, Jonasson and Ranjan Dubhashi, Jonasson and Ranjan (2007) study the negative dependence properties of Srinivasan's sampling processes (SSPs), random processes which sample sets of a fixed size with prescribed marginals. In particular they prove that linear SSPs have conditional negative association, by using the Feder–Mihail theorem and a coupling argument. We consider a broader class of SSPs that we call tournament SSPs (TSSPs). These have a tree-like structure and we prove that they have conditional negative association. Our approach is completely different from that of Dubhashi, Jonasson and Ranjan. We give an abstract characterization of TSSPs, and use this to deduce that certain conditioned TSSPs are themselves TSSPs. We show that TSSPs have negative association, and hence conditional negative association. We also give an example of an SSP that does not have negative association.

Abstract: Let P be a set of n points in ℝ3, and let k ≤ n be an integer. A sphere Iƒ is k-rich with respect to P if |Iƒ ∩ P| ≥ k, and is I·-non-degenerate, for a fixed fraction 0 < I· < 1, if no circle γ ⊂ Iƒ contains more than I·|Iƒ ∩ P| points of P. We improve the previous bound given in [1] on the number of k-rich I·-non-degenerate spheres in 3-space with respect to any set of n points in ℝ3, from O(n4/k5 + n3/k3), which holds for all 0 < I· < 1/2, to O*(n4/k11/2 + n2/k2), which holds for all 0 < I· < 1 (in both bounds, the constants of proportionality depend on I·). The new bound implies the improved upper bound O*(n58/27) ≈ O(n2.1482) on the number of mutually similar triangles spanned by n points in ℝ3; the previous bound was O(n13/6) ≈ O(n2.1667) [1].

Abstract: We show that the number of halving sets of a set of n points in ℝ4 is O(n4−1/18), improving the previous bound of [10] with a simpler (albeit similar) proof.

Abstract: We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does so within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.

Abstract: 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.

Abstract: The classical ErdA‘s-Posa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k + 1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,.i¾ .i¾ .,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class, concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the ErdA‘s-Posa theorem with the extra 'redundancy' property that B-v is still a blocker for all but at most k vertices v ∈ B.

Abstract: We consider a variant of the Cops and Robbers game where the robber can move t edges at a time, and show that in this variant, the cop number of a d-regular graph with girth larger than 2t+2 is I©(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connected n-vertex graph can be as large as I©(n2/3) if t ≥ 2, and I©(n4/5) if t ≥ 4. This improves the I©( $n^{\frac{t-3}{t-2}}$ ) lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers, J. Graph Theory, to appear) when 2 ≤ t ≤ 6. We also conjecture a general upper bound O(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

Abstract: Consider a randomly oriented graph G = (V, E) and let a, s and b be three distinct vertices in V. We study the correlation between the events {a → s} and {s → b}. We show that, counter-intuitively, when G is the complete graph Kn, n ≥ 5, then the correlation is positive. (It is negative for n = 3 and zero for n = 4.) We briefly discuss and pose problems for the same question on other graphs.

Abstract: This article focuses on recent advances and developments of field effect transistor (FET) devices for detecting DNA recognition events such as hybridization, SNP genotyping and primer extension. The unique features of FET biosensors highlight the potential advantages for high-throughput detection of DNA molecules in a label-free manner. In particular, FET devices represent a potential platform for the development of the next-generation DNA sequence instruments based on semiconductor technology. We also review an emerging class of FET devices that use nanomaterials such as silicon nanowires and single-walled carbon nanotubes as a gate channel for the ultrasensitive detection of biological analytes.

Abstract: The line graph G of a directed graph G has a vertex for every edge of G and an edge for every path of length 2 in G. In 1967, Knuth used the Matrix Tree Theorem to prove a formula for the number of spanning trees of G, and he asked for a bijective proof [6]. In this paper, we give a bijective proof of Knuth's formula. As a result of this proof, we find a bijection between binary de Bruijn sequences of degree n and binary sequences of length 2n−1. Finally, we determine the critical groups of all the Kautz graphs and de Bruijn graphs, generalizing a result of Levine [7].

Abstract: Let Ωn be the nn-element set consisting of all functions that have {1, 2, 3,. . ., n} as both domain and codomain. Let T(f) be the order of f, i.e., the period of the sequence f, f(2), f(3), f(4). . . of compositional iterates. A closely related number, B(f) = the product of the lengths of the cycles of f, has previously been used as an approximation for T. This paper proves that the average values of these two quantities are quite different. The expected value of T is \[ \frac{1}{n^{n}}\sum\glimits_{f\in \Omega_{n}}{\bf T}(f)=\exp\biggl(k_{0}\sqrt[3]{\frac{n}{\log^{2}n}}\bigl(1+o(1)\bigr)\biggr), \] where k0 is a complicated but explicitly defined constant that is approximately 3.36. The expected value of B is much larger: \[ \frac{1}{n^{n}}\sum\glimits_{f\in \Omega_{n}}{\bf B}(f)=\exp\biggl(\frac{3}{2}\sqrt[3]{n}\bigl(1+o(1)\bigr)\biggr). \]

Abstract: We show that if G is a simple outerplanar graph and H is a graph with the same Tutte polynomial as G, then H is also outerplanar. Examples show that the condition of G being simple cannot be omitted.