Spanning even subgraphs of 3‐edge‐connected graphs

By Petersen's theorem, a bridgeless cubic graph has a 2‐factor. H. Fleischner extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has a spanning even subgraph. Our main result is that, under the stronger hypothesis of 3‐edge‐connectivity, we can find a spanning even subgraph in which every component has at least five vertices. We show that this is in some sense best possible by constructing an infinite family of 3‐edge‐connected graphs in which every spanning even subgraph has a 5‐cycle as a component. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 37–47, 2009     

The 2‐dimensional rigidity of certain families of graphs

Laman's characterization of minimally rigid 2‐dimensional generic frameworks gives a matroid structure on the edge set of the underlying graph, as was first pointed out and exploited by L. Lovász and Y. Yemini. Global rigidity has only recently been characterized by a combination of two results due to T. Jordán and the first named author, and R. Connelly, respectively. We use these characterizations to investigate how graph theoretic properties such as transitivity, connectivity and regularity influence (2‐dimensional generic) rigidity and global rigidity and apply some of these results to reveal rigidity properties of random graphs. In particular, we characterize the globally rigid vertex transitive graphs, and show that a random d‐regular graph is asymptotically almost surely globally rigid for all d ≥ 4. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 154–166, 2007     

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.     

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3‐edge‐connected graph with and , then either G has an Eulerian subgraph H such that , or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any , G has an Eulerian subgraph H such that . As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003     

Maximum Δ‐edge‐colorable subgraphs of class II graphs

A graph G is class II, if its chromatic index is at least Δ + 1. Let H be a maximum Δ‐edge‐colorable subgraph of G. The paper proves best possible lower bounds for |E(H)|/|E(G)|, and structural properties of maximum Δ‐edge‐colorable subgraphs. It is shown that every set of vertex‐disjoint cycles of a class II graph with Δ≥3 can be extended to a maximum Δ‐edge‐colorable subgraph. Simple graphs have a maximum Δ‐edge‐colorable subgraph such that the complement is a matching. Furthermore, a maximum Δ‐edge‐colorable subgraph of a simple graph is always class I. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

Hamiltonicity and forbidden subgraphs in 4‐connected graphs

Let T be the line graph of the unique tree F on 8 vertices with degree sequence (3,3,3,1,1,1,1,1), i.e., T is a chain of three triangles. We show that every 4‐connected {T, K_1,3}‐free graph has a hamiltonian cycle. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 262–272, 2005     

Four‐dimensional contact ‐submanifolds in

The analogue of ‐submanifolds in (almost) Kählerian manifolds is the concept of contact ‐submanifolds in Sasakian manifolds. These are submanifolds for which the structure vector field ξ is tangent to the submanifold and for which the tangent bundle of M can be decomposed as , where is invariant with respect to the endomorphism φ and is antiinvariant with respect to φ. The lowest possible dimension for M in which this decomposition is non trivial is the dimension 4. In this paper we obtain a complete classification of four‐dimensional contact ‐submanifolds in and for which the second fundamental form restricted to and vanishes identically.     

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

Let M be a Hopf hypersurface in a nonflat complex space form $M^2(c)$, $c\ne 0$, of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical solutions of nonlinear 3‐dimensional Volterra integral‐differential equations with 3D‐block‐pulse functions

This paper studies nonlinear 3‐dimensional Volterra integral‐differential equations, by implementing 3‐dimensional block‐pulse functions. First, we prove a theorem and corollary about sufficient condition for the minimum of mean square error under the block pulse coefficients and uniqueness of solution of the nonlinear Volterra integral‐differential equations. Then, we convert the main problem to a nonlinear system to the 3‐dimensional block‐pulse functions. In addition, illustrative examples are included to demonstrate the validity and applicability of the presented method.     

The links of 3‐dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5‐dimensional links of 3‐dimensional log‐canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ?⁴ such that the order of H₂(K) is 6k. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global well‐posedness of the Cauchy problem for certain magnetohydrodynamic‐α models

This paper is devoted to study the Cauchy problem for certain incompressible magnetohydrodynamics‐α model. In the Sobolev space with fractional index s>1, we proved the local solutions for any initial data, and global solutions for small initial data. Furthermore, we also prove that as α→0, the MHD‐α model reduces to the MHD equations, and the solutions of the MHD‐α model converge to a pair of solutions for the MHD equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Inframonogenic functions and their applications in 3‐dimensional elasticity theory

Solutions of the sandwich equation , where stands for the first‐order differential operator (called Dirac operator) in the Euclidean space , are known as inframonogenic functions. These functions generalize in a natural way the theory of kernels associated with , the nowadays well‐known monogenic functions, and can be viewed also as a refinement of the biharmonic ones. In this paper we deepen study the connections between inframonogenic functions and the solutions of the homogeneous Lamé‐Navier system in . Our findings allow to shed some new light on the structure of the solutions of this fundamental system in 3‐dimensional elasticity theory.     

A three‐dimensional finite element model for the control of certain non‐linear bioreactors

This paper deals with the dynamics of non‐linear distributed parameter fixed‐bed bioreactors. The model consists of a pair of non‐linear partial differential (evolution) equations. The true spatially three‐dimensional situation is considered instead of the usual one‐dimensional approximation. This enables one to take into account the effects of flow profiles and the true location of the measurement transducer. The (output) evolution of the corresponding open‐loop control system is simulated. Furthermore, the associated closed‐loop system with respect to the relevant output function is considered. Especially, the asymptotic output tracking is found to be successful by applying the usual process based on the state feedback linearization. Copyright © 2000 John Wiley & Sons, Ltd.     

On the chromaticity of certain subgraphs of a q-tree

We show that a graph G on n ? q + 1 vertices (where q ? 2) has the chromatic polynomial P(G;λ) = λ(λ ? 1) … (λ ? q + 2) (λ ? q + 1)² (λ ? q)^n?q?1 if and only if G can be obtained from a q-tree Ton n vertices by deleting an edge contained in exactly q ? 1 triangles of T. Furthermore, we prove that these graphs are triangulated.