On an anti‐Ramsey problem of Burr,Erdős,Graham, and T. Sós

Given a graph L, in this article we investigate the anti‐Ramsey number χ_S(n,e,L), defined to be the minimum number of colors needed to edge‐color some graph G(n,e) with n vertices and e edges so that in every copy of L in G all edges have different colors. We call such a copy of L totally multicolored (TMC). In 7 among many other interesting results and problems, Burr, Erd?s, Graham, and T. Sós asked the following question: Let L be a connected bipartite graph which is not a star. Is it true then that In this article, we prove a slightly weaker statement, namely we show that the statement is true if L is a connected bipartite graph, which is not a complete bipartite graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 147–156, 2006     

On bipartite 2‐factorizations of kn − I and the Oberwolfach problem

It is shown that if F₁, F₂, …, F_t are bipartite 2‐regular graphs of order n and α₁, α₂, …, α_t are positive integers such that α₁ + α₂ + ? + α_t = (n ? 2)/2, α₁≥3 is odd, and α_i is even for i = 2, 3, …, t, then there exists a 2‐factorization of K_n ? I in which there are exactly α_i 2‐factors isomorphic to F_i for i = 1, 2, …, t. This result completes the solution of the Oberwolfach problem for bipartite 2‐factors. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:22‐37, 2011     

A note on a third‐order multi‐point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third‐order multi‐point boundary value problem at resonance where f: [0, 1] × R³ → R is a continuous function, 0 < ξ₁ < ??? < ξ_m < 1, α_i ∈ R, i = 1, …, m, m ≥ 1 and 0 < η₁ < η₂ < ??? < η_n < 1, β_j ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′^′′) is equal to 2. Since all the existence results for third‐order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

The Hamilton‐Waterloo problem for Hamilton cycles and triangle‐factors

In this article, we consider the Hamilton‐Waterloo problem for the case of Hamilton cycles and triangle‐factors when the order of the complete graph K_n is even. We completely solved the problem for the case n≡24 (mod 36). For the cases n≡0 (mod 18) and n≡6 (mod 36), we gave an almost complete solution. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 305–316, 2012     

On connectivity,conductance and bootstrap percolation for a random k‐out,age‐biased graph

A uniform attachment graph (with parameter k), denoted G_n,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v ? 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of G_n,k to show that the conductance of G_n,k is of order . We also study the bootstrap percolation on G_n,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.     

The Hamilton–Waterloo problem for cycle sizes 3 and 4

The Hamilton–Waterloo problem seeks a resolvable decomposition of the complete graph K_n, or the complete graph minus a 1‐factor as appropriate, into cycles such that each resolution class contains only cycles of specified sizes. We completely solve the case in which the resolution classes are either all 3‐cycles or 4‐cycles, with a few possible exceptions when n=24 and 48. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 342–352, 2009     

A Hausdorff‐like moment problem and the inversion of the Laplace transform

We consider the problem of finding u ∈ L ²(I ), I = (0, 1), satisfying ∫_I u (x )x dx = μ _k , where k = 0, 1, 2, …, (α _k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ _kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Procrustes problems and inverse eigenproblems for multilevel block α‐circulants

Let n = (n₁,n₂,…,n_k) and α = (α₁,α₂,…,α_k) be integer k‐tuples with α_i∈{1,2,…,n_i?1} and for all i = 1,2,…,k. Multilevel block α ‐circulants are (k + 1)‐level block matrices, where the first k levels have the block α_i‐circulant structure with orders and the entries in the (k + 1)‐st level are unstructured rectangular matrices with the same size . When k = 1, Trench discussed on his paper "Inverse problems for unilevel block α‐circulants" the Procrustes problems and inverse problems of unilevel block α‐circulants and their approximations. But the results are not perfect for the case gcd( α , n ) > 1 (i.e., gcd(α₁,n₁) > 1). In this paper, we also discuss Procrustes problems for multilevel block α ‐circulants. Our results can further make up for the deficiency when k = 1. When , inverse eigenproblems for this kind of matrices are also solved. By using the related results, we can design an artificial Hopfield neural network system that possesses the prescribed equilibria, where the Jacobian matrix of this system has the constrained multilevel α ‐circulative structure. Finally, some examples are employed to illustrate the effectiveness of the proposed results. Copyright © 2016 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     

Counting without sampling: Asymptotics of the log‐partition function for certain statistical physics models

In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard‐core (independent set) model when the activity parameter λ is small, and also the Potts (q‐coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r‐regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4‐regular n‐node graph with large girth has approximately (1.494…)ⁿ‐many independent sets, for large n. Further, we prove that for every r‐regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximately ⁿ, for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well. As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Acyclic 5‐choosability of planar graphs without small cycles

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L‐list colorable if for a given list assignment L = {L(v): v: ∈ V}, there exists a proper acyclic coloring ? of G such that ?(v) ∈ L(v) for all v ∈ V. If G is acyclically L‐list colorable for any list assignment with |L (v)|≥ k for all v ∈ V, then G is acyclically k‐choosable. In this article, we prove that every planar graph G without 4‐ and 5‐cycles, or without 4‐ and 6‐cycles is acyclically 5‐choosable. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 245–260, 2007     

On Hamilton cycles in Erdős‐Rényi subgraphs of large graphs

Given a graph Γ_n=(V,E) on n vertices and m edges, we define the Erd?s‐Rényi graph process with host Γ_n as follows. A permutation e₁,…,e_m of E is chosen uniformly at random, and for t ≤ m we let Γ_n,t=(V,{e₁,…,e_t}). Suppose the minimum degree of Γ_n is δ(Γ_n) ≥ (1/2+ε)n for some constant ε>0. Then with high probability (An event holds with high probability (whp) if as n→∞.), Γ_n,t becomes Hamiltonian at the same moment that its minimum degree reaches 2. Given 0 ≤ p ≤ 1 let Γ_n,p be the Erd?s‐Rényi subgraph of Γ_n, obtained by retaining each edge independently with probability p. When δ(Γ_n) ≥ (1/2+ε)n, we provide a threshold for Hamiltonicity in Γ_n,p.     

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

An n × n real matrix A = (a_ij)_{n × n} is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, a_ij = a_ji and a_ij = a_n+1?1,n+1?i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A^* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

Sparse universal graphs for bounded‐degree graphs

Let ℋ︁ be a family of graphs. A graph T is ℋ︁‐universal if it contains a copy of each H ∈ℋ︁ as a subgraph. Let ℋ︁(k,n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an ℋ︁(k,n)‐universal graph T with edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n^2‐2/k) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Minors in large almost‐5‐connected non‐planar graphs

It is shown that every sufficiently large almost‐5‐connected non‐planar graph contains a minor isomorphic to an arbitrarily large graph from one of six families of graphs. The graphs in these families are also almost‐5‐connected, by which we mean that they are 4‐connected and all 4‐separations contain a “small” side. As a corollary, every sufficiently large almost‐5‐connected non‐planar graph contains both a K_{3, 4}‐minor and a ‐minor. The connectivity condition cannot be reduced to 4‐connectivity, as there are known infinite families of 4‐connected non‐planar graphs that do not contain a K_{3, 4}‐minor. Similarly, there are known infinite families of 4‐connected non‐planar graphs that do not contain a ‐minor.     

Disproof of a packing conjecture of Alon and Spencer

A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph G_n,1/2 typically admits a covering of a constant fraction of its edges by edge‐disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: for and A₁,…,A_t chosen uniformly and independently from the k‐subsets of {1,…,n}, what can one say about Our main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently.     

Rank‐width of random graphs

Rank‐width of a graph G, denoted by rw (G), is a width parameter of graphs introduced by Oum and Seymour [J Combin Theory Ser B 96 (2006), 514–528]. We investigate the asymptotic behavior of rank‐width of a random graph G(n, p). We show that, asymptotically almost surely, (i) if p∈(0, 1) is a constant, then rw (G(n, p)) = ?n/3??O(1), (ii) if , then rw (G(n, p)) = ?1/3??o(n), (iii) if p = c/n and c>1, then rw (G(n, p))?rn for some r = r(c), and (iv) if p?c/n and c81, then rw (G(n, p))?2. As a corollary, we deduce that the tree‐width of G(n, p) is linear in n whenever p = c/n for each c>1, answering a question of Gao [2006]. © 2011 Wiley Periodicals, Inc. J Graph Theory.     

Complete solutions to the Oberwolfach problem for an infinite set of orders

Let n3 and let F be a 2-regular graph of order n. The Oberwolfach problem OP(F) asks for a 2-factorisation of K_n if n is odd, or of K_n−I if n is even, in which each 2-factor is isomorphic to F. We show that there is an infinite set of primes congruent to such that OP(F) has a solution for any 2-regular graph F of order . We also show that for each of the infinitely many with prime, OP(F) has a solution for any 2-regular graph F of order n.