The A4‐structure of a graph

We define the A₄‐structure of a graph G to be the 4‐uniform hypergraph on the vertex set of G whose edges are the vertex subsets inducing 2K₂, C₄, or P₄. We show that perfection of a graph is determined by its A₄‐structure. We relate the A₄‐structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Math 220 (2000) 201–238]; for example, a graph is indecomposable if and only if its A₄‐structure is connected. We also characterize the graphs having the same A₄‐structure as a split graph.     

3‐colorability of 4‐regular hamiltonian graphs

On the model of the cycle‐plus‐triangles theorem, we consider the problem of 3‐colorability of those 4‐regular hamiltonian graphs for which the components of the edge‐complement of a given hamiltonian cycle are non‐selfcrossing cycles of constant length ≥ 4. We show that this problem is NP‐complete. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 125–140, 2003     

6‐regular 4‐critical graph

P. Erd?s conjectured in [2] that r‐regular 4‐critical graphs exist for every r ≥ 3 and noted that no such graphs are known for r ≥ 6. This article contains the first example of a 6‐regular 4‐critical graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 286–291, 2002     

Existence of magic 3‐dimensional rectangles

Since the complete solution for the existence of magic 2‐dimensional rectangles in 1881, much attention has been paid on the existence of magic l‐dimensional rectangles for . The existence problem for magic l‐dimensional rectangles with even sizes has been solved completely for all integers . However, very little is known for the existence of magic l‐dimensional rectangles () with odd sizes except for some families and a few sporadic examples. In this paper, we focus our attention on the existence of magic 3‐dimensional rectangles and prove that the necessary conditions for the existence of magic 3‐dimensional rectangles are also sufficient. Our construction method is mainly based on a new concept, symmetric zero‐sum subset partition, which plays a crucial role in the recursive constructions of magic 3‐rectangles similar to that of PBD in the PBD‐closure construction in combinatorial design theory.     

Symmetric and Skew‐Symmetric ‐Matrices with Large Determinants

We show that the existence of ‐matrices having largest possible determinant is equivalent to the existence of certain tournament matrices. In particular, we prove a recent conjecture of Armario. We also show that large submatrices of conference matrices are determined by their spectrum.     

4‐*GDD(6n)s and Related Optimal Quaternary Constant‐Weight Codes

Constant‐weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a K‐GDD (where GDD is group divisible design) with the “star” property, denoted by K‐*GDD, was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4‐*GDDs. Previously, the necessary conditions for existence were shown to be sufficient for , and also sufficient for with prime powers and . We continue to investigate the existence of 4‐*GDD(6ⁿ)s and show that the necessary condition for the existence of a 4‐*GDD(6ⁿ), namely, , is also sufficient. The known results on the existence of optimal quaternary (n, 5, 4) CWCs are also extended.     

Algebraic multigrid and 4th‐order discrete‐difference equations of incompressible fluid flow

This paper investigates the effectiveness of two different Algebraic Multigrid (AMG) approaches to the solution of 4th‐order discrete‐difference equations for incompressible fluid flow (in this case for a discrete, scalar, stream‐function field). One is based on a classical, algebraic multigrid, method (C‐AMG) the other is based on a smoothed‐aggregation method for 4th‐order problems (SA‐AMG). In the C‐AMG case, the inter‐grid transfer operators are enhanced using Jacobi relaxation. In the SA‐AMG case, they are improved using a constrained energy optimization of the coarse‐grid basis functions. Both approaches are shown to be effective for discretizations based on uniform, structured and unstructured, meshes. They both give good convergence factors that are largely independent of the mesh size/bandwidth. The SA‐AMG approach, however, is more costly both in storage and operations. The Jacobi‐relaxed C‐AMG approach is faster, by a factor of between 2 and 4 for two‐dimensional problems, even though its reduction factors are inferior to those of SA‐AMG. For non‐uniform meshes, the accuracy of this particular discretization degrades from 2nd to 1st order and the convergence factors for both methods then become mesh dependent. Copyright © 2009 John Wiley & Sons, Ltd.     

1‐rotationally resolvable 4‐cycle systems of 2Kv

In this article, it is shown that there exists a 1‐rotationally resolvable 4‐cycle system of 2K_υ if and only if υ ≡ 0 (mod 4). To prove that, some special sequences of integers are utilized. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 116–125, 2002; DOI 10.1002/jcd.10006     

Symmetric ‐Designs and ‐Difference Sets

In this paper, we introduce a method to construct ‐designs, which are also known as partial geometric designs, by using subsets of certain finite groups. We introduce the concept of ‐difference sets and investigate the existence and nonexistence of these structures. We also provide some nonexistence results on ‐designs based on the fact that ‐designs yield directed strongly regular graphs.     

Existence and zero‐electron‐mass limit of strong solutions to the stationary compressible Navier‐Stokes‐Poisson equation with large external force

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier‐Stokes‐Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero‐electron‐mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier‐Stokes equations with large forces, a system of stationary compressible Navier‐Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier‐Stokes equation, linear compressible Navier‐Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero‐electron‐mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier‐Stokes‐Poisson equations.     

Weak solution to a one‐dimensional full compressible non‐Newtonian fluid

This paper is devoted to the existence of global‐in‐time weak solutions to a one‐dimensional full compressible non‐Newtonian fluid. A semi‐discrete finite element scheme is taken to generate approximate solutions, based on an exact projection technique. To enforce convergence of the approximate solutions, the uniform estimate is obtained using an iteration method and energy method, with the help of the weak compactness and convexity. Numerical simulations showing the existence of solutions are presented.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

L∞‐solutions for the ultra‐slow reaction‐diffusion equation

The existence and uniqueness of solutions for a reaction‐diffusion ultra‐slow equation are proved. We also show that they can be extended up a maximal time and are stable as long as they exist. Symmetric and positive solutions are also proved to exist.     

Some more 5‐GDDs and optimal (v), 5, 1)‐packings

In this paper, we continue to investigate the existence of 5‐GDDs with types g⁵ m¹ and g^u. As an application, the known results on the existence of optimal packings with block size 5 are also improved. © 2003 Wiley Periodicals, Inc.     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

Uniform domain representations of ℓp ‐spaces

The category of Scott‐domains gives a computability theory for possibly uncountable topological spaces, via representations. In particular, every separable Banach‐space is representable over a separable domain. A large class of topological spaces, including all Banach‐spaces, is representable by domains, and in domain theory, there is a well‐understood notion of parametrizations over a domain. We explore the link with parameter‐dependent collections of spaces in e. g. functional analysis through a case study of ?^p ‐spaces. We show that a well‐known domain representation of ?^p as a metric space can be made uniform in the sense of parametrizations of domains. The uniform representations admit lifting of continuous functions and are effective in p. Dependent type constructions apply, and through the study of the sum and product spaces, we clarify the notions of uniformity and uniform computability. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The existence of a 2‐factor in K1, n‐free graphs with large connectivity and large edge‐connectivity

In this article, we study the existence of a 2‐factor in a K_{1, n}‐free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n?4, an (n?1)‐connected K_{1, n}‐free graph of even order has a 1‐factor. On the other hand, for every pair of integers m and n with m?n?4, there exist infinitely many (n?2)‐connected K_{1, n}‐free graphs of even order and minimum degree at least m which have no 1‐factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1‐factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n?3, every K_{1, n}‐free graph of minimum degree at least 2n?2 has a 2‐factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge‐connectivity and minimum degree to the existence of a 2‐factor in a K_{1, n}‐free graph are more complicated than those to the existence of a 1‐factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K_{1, n}‐free graph with a given connectivity or edge‐connectivity to have a 2‐factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77‐89, 2011     

Some new results on 1‐rotational 2‐factorizations of the complete graph

It is known that a necessary condition for the existence of a 1‐rotational 2‐factorization of the complete graph K_2n+1 under the action of a group G of order 2n is that the involutions of G are pairwise conjugate. Is this condition also sufficient? The complete answer is still unknown. Adapting the composition technique shown in Buratti and Rinaldi, J Combin Des, 16 (2008), 87–100, we give a positive answer for new classes of groups; for example, the groups G whose involutions lie in the same conjugacy class and having a normal subgroup whose order is the greatest odd divisor of |G|. In particular, every group of order 4t+2 gives a positive answer. Finally, we show that such a composition technique provides 2‐factorizations with a rich group of automorphisms. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 237–247, 2010     

Cyclic 4‐Colorings of Graphs on Surfaces

To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4‐vertex‐coloring: a plane triangulation G has a proper 4‐vertex‐coloring if and only if the dual of G has a proper 3‐edge‐coloring. A cyclic coloring of a map G on a surface F² is a vertex‐coloring of G such that any two vertices x and y receive different colors if x and y are incident with a common face of G. In this article, we extend the result by Tait to two directions, that is, considering maps on a nonspherical surface and cyclic 4‐colorings.