Abstract: We consider linear preferential attachment trees, and show that they can be regarded as random split trees in the sense of Devroye (1999), although with infinite potential branching. In particular, this applies to the random recursive tree and the standard preferential attachment tree. An application is given to the sum over all pairs of nodes of the common number of ancestors.

Abstract: For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $ H $ contains a monochromatic copy of $G_i$ of color $i$ for some $1\leq i\leq k$. We denote $\hat{R}(G_1,\ldots,G_k)$ by $\hat{R}_{k}(G)$ when $G_1=\cdots=G_k=G$. Haxell, Kohayakawa and \L{}uczak showed that the size-Ramsey number of a cycle $C_n$ is linear in $n$ i.e. $\hat{R}_{k}(C_{n})\leq c_k n$ for some constant $c_k$. Their proof, however, is based on the regularity lemma of Szemer\'{e}di and so no specific constant $c_k$ is known. In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of $\hat{R}_{k}(C_{n})\leq c_k n$, avoiding the use of the regularity lemma, where $ c_k $ is exponential and doubly-exponential in $ k $, when $ n $ is even and odd, respectively. In particular, we show that for sufficiently large $n$ we have $\hat{R}(C_{n},C_{n}) \leq 10^5\times cn,$ where $c=6.5$ if $n$ is even and $c=1989$ otherwise.

Abstract: Fluorescence based bioimaging is one of the widely used method for obtaining imperative information on life processes. Within the expansive spectrum of fluorescent agents being investigated, the trivalent Lanthanide (Ln) ion based nanoparticles have attracted attention due to their intrinsic luminescence property. Here we report a modified sol gel assisted synthesis of Europium (Eu) and Samarium (Sm) doped Hydroxyapatite nanoparticles (HAp NPs). Doping Ln ions in the selffluorescent hydroxyapatite lattice contributed towards an increased luminescence in the NPs. The XRD patterns reveal that the Eu+3 and Sm+3 doped HAp NPs display the characteristic peaks of hydroxyapatite in a hexagonal lattice structure, and the FTIR data confirms presence of characteristic functional groups. The as-synthesized HAp NPs exhibit short rod-shaped morphology with average length less than 60 nm. Upon excitation at representative wavelengths, the doped HAp NPs demonstrated characteristic emission lines of Eu+3 and Sm+3. The as-synthesized NPs displayed no toxicity towards HeLa cells and are easily internalized, exhibiting their potential as promising live cell bioimaging agents.

Abstract: We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

Abstract: The hard-core model has attracted much attention across several disciplines, representing lattice gases in statistical physics and independent sets in discrete mathematics and computer science. On finite graphs, we are given a parameter λ, and an independent set I arises with probability proportional to λ|I|. On infinite graphs a Gibbs measure is defined as a suitable limit with the correct conditional probabilities, and we are interested in determining when this limit is unique and when there is phase coexistence, i.e., existence of multiple Gibbs measures.It has long been conjectured that on ℤ2 this model has a critical value λc ≈ 3.796 with the property that if λ λc then there is phase coexistence. Much of the work to date on this problem has focused on the regime of uniqueness, with the state of the art being recent work of Sinclair, Srivastava, Stefankovic and Yin showing that there is a unique Gibbs measure for all λ 5.3506. We also show that with the methods we are using we cannot hope to replace 5.3506 with anything below 4.8771.Our proof begins along the lines of the standard Peierls argument, but we add two innovations. First, following ideas of Kotecký and Randall, we construct an event that distinguishes two boundary conditions and always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of ℤ2.

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

Abstract: Let r ⩾ 2 be a fixed constant and let $ {\cal H} $ be an r -uniform, D -regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that degrees of pairs of vertices in $ {\cal H} $ are at most L where L = D/ ( log N ) ω (1) . We consider the random greedy algorithm for forming a matching in $ {\cal H} $ . We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\cal H} $ that are not saturated by the final matching is at most ( L / D ) (1/(2( r −1)))+ o (1) . This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

Abstract: For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

Abstract: We prove that the number of multigraphs with vertex set {1, . . ., n} such that every four vertices span at most nine edges is an2+o(n2) where a is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments.Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

Abstract: A family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any , in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

Abstract: For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.

Abstract: An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a sub ...

Abstract: Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

Abstract: Recently, great importance has been devoted to borate glass systems doped with rare-earth ions because of their unique peculiar properties in the field of photonics for optical applications. The purpose of the present study is to investigate the effect of concentration of Sm3+ ions on the luminescence properties of lead fluoroborate glasses through the energy transfer mechanism. Samarium doped lead fluoroborate glasses with chemical composition 20PbF2 .10Li2O .5SrO .5ZnO. (60-x) B2O3. xSm2O3 (where x = 0.1, 0.5, 1.0, 1.5 and 2.0 mol %) were prepared by means of melt quenching method. The concentration dependent luminescence properties were investigated in detail from the optical absorption, photoluminescence and decay analysis. Judd-Ofelt (J-O) theory was applied to analyze the optical absorption spectra. The experimental oscillator strengths of absorption bands have been used to determine the J-O parameters. Using the J-O parameters Ωλ (λ = 2, 4 and 6) and luminescence data several radiative parameters were obtained. From the luminescence spectra, it was noticed that luminescence quenching starts at higher concentrations of Sm3+ ions (x ≥ 0.5 mol %). The decay curves of 4G5/2 → 6H7/2 transition exhibit a single exponential at lower dopant concentrations (x= 0.1 and 0.5 mol %) and non-exponential at higher concentrations (x ≥ 1 mol %). The concentration quenching was attributed to the energy transfer through the cross-relaxation between Sm3+ ions. The non-exponential curves were well fitted to Inokuti-Hirayama model for S = 6, indicating that the energy transfer between Sm3+ - Sm3+ ions is of dipole-dipole type. The calculated color coordinates of the as-prepared glasses fall within the reddish-orange region of the CIE diagram. All the experimental results indicate that the 0.5 mol% Sm3+ ions doped LLSZFB glass can be a possible choice for solid state lighting and display applications.

Abstract: A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

Abstract: The azadirachtin is a triterpenoid associated with growth inhibition in several kinds of insects which cause epidemic diseases like Dengue, Chikungunya and Malaria. Azadirachtin acts by inhibiting the Ecdysone Receptor (EcR), which is responsible from larvae phase in insects. However, the interaction between the azadirachtin molecule and the Ecdysone Receptor is unknown. In this work, we used the program Dock Thor to generate several azadirachtin conformations inside the EcR binding site. The ten most stable conformations were optimized with the ONIOM approach present in the Gaussian 09 program. The interaction energy was calculated between the azadirachtin molecule and EcR receptor. Theoretical calculation shows that the azadirachtin molecule interacts with the same amino acids present in the ecdysone EcR interaction. These results will be useful to design new EcR inhibitors, which can be used in the control of some diseases based on insect proliferations. To understand the interaction between the natural insecticide azadirachtin and the Ecdysone Receptor. A combination of Dock Thor program with QM-MM calculation was used in order to obtain the most favorable molecular structures. The hydrogens bond obtained by Dock Thor Program combined with QM-MM calculation suggest the azadirachtin interact with EcR in the same way that ecdysone molecule. The interaction mode that the molecule azadirachtin inhibits EcR in order to avoid insect proliferation was described.

Abstract: To synthesize biocompatible nanoparticles of FAp co-doped with Yb/Er and Nd/Yb for bioimaging applications. Yb/Er FAp and Nd/Yb FAp was synthesized using microwave assisted wet precipitation and hydrothermal method respectively. Trisodium citrate was used as an organic modifier for the synthesis and then subjected to heat treatment for optical activation. For optical studies, Yb/Er FAp system was excited at 980 nm and Nd/Yb FAp at 800 nm. In the case of Nd/Yb FAp the host matrix absorption and emission was observed, hence Nd/Yb was synthesized without citrate. On heat treatment of this for optical activation studies, when the Yb3+ concentration was increased to 10 mol%, the YbPO4 secondary phase was found to appear. Although, the Yb/Er FAp system resulted in large grain growth, no such grain growth was observed in Nd/Yb FAp and the grains were within the nano size regime even after heat treatment. Both the systems showed successful energy transfer from sensitizer to activator with a quantum yield of 74% for Yb/Er FAp and energy transfer efficiency of 71% for Nd/Yb FAp system. Both the samples were found to be cytocompatible and has the potential for using as probes for bioimaging applications.

Abstract: An (improper) graph colouring has defect d if each monochromatic subgraph has maximum degree at most d, and has clustering c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degree m, no (1-ɛ)m bound on the number of colours was previously known. The above result with d=1 solves this problem. It implies that every graph with maximum average degree m is -choosable with clustering O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.

Abstract: We present an average-case analysis of a variant of dual-pivot quicksort. We show that the algorithmic partitioning strategy used is optimal, that is, it minimizes the expected number of key comparisons. For the analysis, we calculate the expected number of comparisons exactly as well as asymptotically; in particular, we provide exact expressions for the linear, logarithmic and constant terms.An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proved.

Abstract: There was an incorrect argument in the proof of the main theorem in ‘On percolation and the bunkbed conjecture’, in Combin. Probab. Comput. (2011) 20 103–117 doi: 10.1017/S0963548309990666. I thus no longer claim to have a proof for the bunkbed conjecture for outerplanar graphs.

Abstract: We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbours coloured by 1, and locally stable functions, in which every vertex has a constant fraction of neighbours coloured by its own colour. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for ℤn and other graphs, and offer several follow-up questions.

Abstract: Given two k-graphs (k-uniform hypergraphs) F and H, a perfect F-tiling (or F-factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. For all complete k-partite k-graphs K, Mycroft proved a minimum codegree condition that guarantees a K-factor in an n-vertex k-graph, which is tight up to an error term o(n). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turan number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K(k)(1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.

Abstract: We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

Abstract: The strong chromatic number χs(G) of a graph G on n vertices is the least number r with the following property: after adding isolated vertices to G and taking the union with any collection of spanning disjoint copies of Kr in the same vertex set, the resulting graph has a proper vertex colouring with r colours. We show that for every c > 0 and every graph G on n vertices with Δ(G) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

Abstract: We consider the asymptotics of the difference between the empirical measures of the β-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik–Kerov–Logan–Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.