On the density of 2‐colorable 3‐graphs in which any four points span at most two edges

Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)$. We improve the previously best known lower and upper bounds of 0.25682 and 3/10?ε, respectively, by showing that This implies the following new upper bound for the Turán density of K In order to establish these results we use a combination of the properties of computer‐generated extremal 3‐graphs for small n and an argument based on “super‐saturation”. Our computer results determine the exact values of ex(n, K) for n≤19 and ex₂(n, K) for n≤17, as well as the sets of extremal 3‐graphs for those n. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 105–114, 2010     

Note on a problem of M. Talagrand

Answering a question of M. Talagrand, we show that there is a fixed L with the following property. For positive integers and , if is the set of subgraphs of K_n containing at least copies of K_k, then there is a set of subgraphs of K_n such that (i) each member of contains a member of and (ii) (where means number of edges). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 663–668, 2015     

Rainbow matchings and Hamilton cycles in random graphs

Let be drawn uniformly from all m‐edge, k‐uniform, k‐partite hypergraphs where each part of the partition is a disjoint copy of . We let be an edge colored version, where we color each edge randomly from one of colors. We show that if and where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if n is even and where K is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in . Here denotes a random edge coloring of with n colors. When n is odd, our proof requires for there to be a rainbow Hamilton cycle. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 503–523, 2016     

Four random permutations conjugated by an adversary generate Sn with high probability

We prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group S_n generate a transitive subgroup with probability for some independent of n, even when an adversary is allowed to conjugate each of the four by a possibly different element of . In other words, the cycle types already guarantee generation of a transitive subgroup; by a well known argument, this implies generation of A_n or except for probability as . The analysis is closely related to the following random set model. A random set is generated by including each independently with probability . The sumset is formed. Then at most four independent copies of are needed before their mutual intersection is no longer infinite. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 409–428, 2016     

On Ramsey‐type positional games

Let K denote the graph obtained from the complete graph K_s+t by deleting the edges of some K_t‐subgraph. We prove that for each fixed s and sufficiently large t, every graph with chromatic number s+t has a K minor. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 343–350, 2010     

On induced Ramsey numbers for k‐uniform hypergraphs

Let denote the complete k‐uniform k‐partite hypergraph with classes of size t and the complete k‐uniform hypergraph of order s. One can show that the Ramsey number for and satisfies when t = s^o(1) as s →∞. The main part of this paper gives an analogous result for induced Ramsey numbers: Let be an arbitrary k‐partite k‐uniform hypergraph with classes of size t and an arbitrary k‐graph of order s. We use the probabilistic method to show that the induced Ramsey number (i.e. the smallest n for which there exists a hypergraph such that any red/blue coloring of yields either an induced red copy of or an induced blue copy of ) satisfies . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 5–20, 2016     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cutting two graphs simultaneously

Consider two graphs, and , on the same vertex set V, with and having edges for . We give a simple algorithm that partitions V into sets A and B such that and . We also show, using a probabilistic method, that if and belong to certain classes of graphs, (for instance, if and both have a density of at least 2/, or if and are both regular of degree at most with n sufficiently large) then we can find a partition of V into sets A and B such that for . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 19–32, 2008     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

Note on the iterative methods for computing the generalized inverse over Banach spaces

In this paper, we construct iterative methods for computing the generalized inverse A over Banach spaces, and also for computing the generalized Drazin inverses a^d of Banach algebra element a. Moreover, we estimate the error bounds of the iterative methods for approximating A or a^d. Copyright © 2011 John Wiley & Sons, Ltd.     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008     

Clairvoyant embedding in one dimension

Let v, w be infinite 0‐1 sequences, and a positive integer. We say that is ‐embeddable in , if there exists an increasing sequence of integers with , such that , for all . Let and be coin‐tossing sequences. We will show that there is an with the property that is ‐embeddable into with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 520–560, 2015     

Maximum number of colorings of (2k,k2)‐graphs

Let consist of all simple graphs on 2k vertices and edges. For a simple graph G and a positive integer , let denote the number of proper vertex colorings of G in at most colors, and let . We prove that and is the only extremal graph. We also prove that as . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 135–148, 2007     

Compatible Hamilton cycles in random graphs

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability , the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph , an incompatibility system over G is a family where for every , the set F_v is a set of unordered pairs . An incompatibility system is Δ‐bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in F_v containing e. We say that a cycle C in G is compatible with if every pair of incident edges of C satisfies . This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant such that the random graph with is asymptotically almost surely such that for any μnp‐bounded incompatibility system over G, there is a Hamilton cycle in G compatible with . We also prove that for larger edge probabilities , the parameter μ can be taken to be any constant smaller than . These results imply in particular that typically in G(n, p) for , for any edge‐coloring in which each color appears at most μnp times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge‐coloring of such a random graph in which each color appears on at most μnp edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 533–557, 2016     

Robust characterizations of k‐wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non‐uniform) k ‐wise independent if for any subset of k indices {i₁,…,i_k} and for any z₁∈Σ,…,z_k∈Σ, Pr_X～D[X···X = z₁···z_k] = Pr_X～D[X = z₁]···Pr_X～D[X = z_k]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Circular chromatic index of Cartesian products of graphs

The circular chromatic index of a graph G, written , is the minimum r permitting a function such that whenever e and are incident. Let □ , where □ denotes Cartesian product and H is an ‐regular graph of odd order, with (thus, G is s‐regular). We prove that , where is the minimum, over all bases of the cycle space of H, of the maximum length of a cycle in the basis. When and m is large, the lower bound is sharp. In particular, if , then □ , independent of m. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 7–18, 2008     

On k‐detour subgraphs of hypercubes

A spanning subgraph G of a graph H is a k‐detour subgraph of H if for each pair of vertices , the distance, , between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this article, we study k‐detour subgraphs of the n‐dimensional cube, , with few edges or with moderate maximum degree. Let denote the minimum possible maximum degree of a k‐detour subgraph of . The main result is that for every and On the other hand, for each fixed even and large n, there exists a k‐detour subgraph of with average degree at most . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 55–64, 2008     

A note on complete subdivisions in digraphs of large outdegree

Mader conjectured that for all there is an integer such that every digraph of minimum outdegree at least contains a subdivision of a transitive tournament of order . In this note, we observe that if the minimum outdegree of a digraph is sufficiently large compared to its order then one can even guarantee a subdivision of a large complete digraph. More precisely, let be a digraph of order n whose minimum outdegree is at least d. Then contains a subdivision of a complete digraph of order . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 1–6, 2008     

Randomized rumor spreading in poorly connected small‐world networks

The Push‐Pull protocol is a well‐studied round‐robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread it to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze the behavior of this protocol on random ‐trees, a class of power law graphs, which are small‐world and have large clustering coefficients, built as follows: initially we have a ‐clique. In every step a new node is born, a random ‐clique of the current graph is chosen, and the new node is joined to all nodes of the ‐clique. When is fixed, we show that if initially a random node is aware of the rumor, then with probability after rounds the rumor propagates to nodes, where is the number of nodes and is any slowly growing function. Since these graphs have polynomially small conductance, vertex expansion and constant treewidth, these results demonstrate that Push‐Pull can be efficient even on poorly connected networks. On the negative side, we prove that with probability the protocol needs at least rounds to inform all nodes. This exponential dichotomy between time required for informing almost all and all nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound successfully carries over to a closely related class of graphs, the random ‐Apollonian networks, for which we prove an upper bound of rounds for informing nodes with probability when is fixed. Here, © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 185–208, 2016     

Vertex suppression in 3‐connected graphs

To suppress a vertex in a finite graph G means to delete it and add an edge from a to b if a, b are distinct nonadjacent vertices which formed the neighborhood of . Let be the graph obtained from by suppressing vertices of degree at most 2 as long as it is possible; this is proven to be well defined. Our main result states that every 3‐connected graph G has a vertex x such that is 3‐connected unless G is isomorphic to , , or to a wheel for some . This leads to a generator theorem for 3‐connected graphs in terms of series parallel extensions. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 41–54, 2008