Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing‐critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k‐crossing‐critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k‐crossing‐critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003     

Minors in graphs of large girth

We show that for every odd integer g ≥ 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr^(g+1)/4. This is best possible up to the value of the constant c for g = 5, 7, and 11. More generally, a well‐known conjecture about the minimal order of graphs of given minimum degree and large girth would imply that our result gives the correct order of magnitude for all odd values of g. The case g = 5 of our result implies Hadwiger's conjecture for C₄‐free graphs of sufficiently large chromatic number. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 213–225, 2003     

2‐connected 7‐coverings of 3‐connected graphs on surfaces

An m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3‐connected graph G on F² with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003     

A scaling limit for the degree distribution in sublinear preferential attachment schemes

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to ‘non‐tree’ evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C₀‐semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 703–731, 2016     

A minimal pair joining to a plus cupping Turing degree

A computably enumerable (c.e.) degree a is called nonbounding, if it bounds no minimal pair, and plus cupping, if every nonzero c.e. degree x below a is cuppable. Let NB and PC be the sets of all nonbounding and plus cupping c.e. degrees, respectively. Both NB and PC are well understood, but it has not been possible so far to distinguish between the two classes. In the present paper, we investigate the relationship between the classes NB and PC , and show that there exists a minimal pair which join to a plus cupping degree, so that PC ? NB . This gives a first known difference between NB and PC . (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polynomial time approximation schemes for dense instances of minimum constraint satisfaction

It is known that large fragments of the class of dense Minimum Constraint Satisfaction (MIN‐CSP) problems do not have polynomial time approximation schemes (PTASs) contrary to their Maximum Constraint Satisfaction analogs. In this paper we prove, somewhat surprisingly, that the minimum satisfaction of dense instances of kSAT ‐formulas, and linear equations mod 2, Ek‐LIN2, do have PTASs for any k. The MIN‐Ek‐LIN2 problems are equivalent to the k‐ary versions of the Nearest Codeword problem, the problem which is known to be exceedingly hard to approximate on general instances. The method of solution of the above problems depends on the development of a new density sampling technique for k‐uniform hypergraphs which could be of independent interest. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 73–91, 2003     

Some New uniform frames with block size four and index one or three

In this paper, some new uniform frames with block size four and index one or three are obtained. The known existence results for (4,1)‐frames and (4,3)‐frames are both improved to the extent that only a finite number of possible exceptions remain. © 2003 Wiley Periodicals, Inc.     

An algorithm for finding factorizations of complete graphs

We show how to find a decomposition of the edge set of the complete graph into regular factors where the degree and edge‐connectivity of each factor is prescribed. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 132–136, 2003     

Resolvable even‐cycle systems with a 1‐rotational automorphism

In this article, necessary and sufficient conditions for the existence of a 1‐rotationally resolvable even‐cycle system of λK_v are given, which are eventually for the existence of a resolvable even‐cycle system of λK_v. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 394–407, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10058     

Investigations of nonstandard,Mickens‐type,finite‐difference schemes for singular boundary value problems in cylindrical or spherical coordinates

It is well known that standard finite‐difference schemes for singular boundary value problems involving the Laplacian have difficulty capturing the singular (??(1/r) or ??(log r)) behavior of the solution near the origin (r = 0). New nonstandard finite‐difference schemes that can capture this behavior exactly for certain singular boundary value problems encountered in theoretical aerodynamics are presented here. These schemes are special cases of nonstandard finite differences which have been extensively researched by Professor Ronald E. Mickens of Clark Atlanta University in their most general form. Several examples of these “Mickens‐type” finite differences that illustrate both their accuracy and utility for singular boundary value problems in both cylindrical and spherical co‐ordinates are investigated. The numerical results generated by the Mickens‐type schemes are compared favorably with solutions obtained from standard finite‐difference schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 380–398, 2003.     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

On Moore bipartite digraphs

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore‐like bound in terms of its diameter k and the maximum out‐degrees (d₁, d₂) of its partite sets of vertices. It has been proved that, when d₁d₂ > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 ≤ k ≤ 4. This paper deals with the problem of their enumeration. In this context, using the theory of circulant matrices and the so‐called De Bruijn near‐factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out‐degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k = 4, which means that G = L G′, where G′ is a Moore bipartite digraph of diameter k = 3. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 171–187, 2003     

Quasi‐complements of the cappable degrees

Say that a nonzero c. e. degree b is a quasi‐complement of a c. e. degree a if a ∩ b = 0 and a ∪ b is high. It is well‐known (due to Shore) that each cappable degree has a high quasi‐complement. However, by the existence of the almost deep degrees, there are nonzero cappable degrees having no low quasi‐complements. In this paper, we prove that any nonzero cappable degree has a low₂ quasi‐complement. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Halley-type iterations with free second derivative

In this paper, we relax the convergence conditions required in Ezquerro and Hernández (Int. J. Pure Appl. Math. 6(1) (2003) 103) for a multipoint third-order iteration of Halley type, where the conditions provided are the known ones for methods of order three.     

Existence and high regularity of the solution of a nonlinear parabolic problem with algebraic‐differential boundary conditions

In this paper we investigate a nonlinear 1D parabolic problem with algebraic‐differential boundary conditions. Existence, uniqueness and higher regularity of the solution is proved. It is shown that actually any regularity can be obtained provided that appropriate smoothness of the data and compatibility assumptions are required. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the mixed finite element method with Lagrange multipliers

In this note we analyze a modified mixed finite element method for second‐order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R ². The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu?ka‐Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart‐Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192–210, 2003     

Existence and global exponential stability of anti‐periodic solutions for delayed quaternion‐valued cellular neural networks with impulsive effects

In this paper, we consider a class of delayed quaternion‐valued cellular neural networks (DQVCNNs) with impulsive effects. By using a novel continuation theorem of coincidence degree theory, the existence of anti‐periodic solutions for DQVCNNs is obtained with or without assuming that the activation functions are bounded. Furthermore, by constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of anti‐periodic solutions for DQVCNNs. Our results are new and complementary to the known results even when DQVCNNs degenerate into real‐valued or complex‐valued neural networks. Finally, an example is given to illustrate the effectiveness of the obtained results.     

Vertex‐distinguishing edge colorings of graphs

We consider lower bounds on the the vertex‐distinguishing edge chromatic number of graphs and prove that these are compatible with a conjecture of Burris and Schelp 8 . We also find upper bounds on this number for certain regular graphs G of low degree and hence verify the conjecture for a reasonably large class of such graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 95–109, 2003     

Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case

This article studies the stability and convergence of the hp version of the three families of mixed discontinuous finite element (MDFE) methods for the numerical solution of reaction‐diffusion problems. The focus of this article is on these problems for one space dimension. Error estimates are obtained explicitly in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Extensive numerical results to show convergence rates in h and p of the MDFE methods are presented. Theoretical and numerical comparisons between the three families of MDFE methods are described. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 525–553, 2003