PROPERTIES OF FRACTIONAL k-FACTORS OF GRAPHS

In this paper the properties of some maximum fractional [0, k]-factors of graphs are presented. And consequently some results on fractional matchings and fractional 1-factors are generalized and a characterization of fractional k-factors is obtained.     

EXTENSIONS OF SOME THEOREMS ON 1-FACTORS

Let G be a graph and n be a positive integer. A spanning subgraph F of G is called a {1, 3, …, 2n-1} -factor if d_F(x)∈{1, 3, …, 2n-1} for all x∈ V(G). Here we give several results on {1, 3, …, 2n-1} -factors, which are the extensions of some theorems on 1-factors given by Las Vergnas, Sumner and others.     

EXISTENCE OF POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, using the property of the corresponding Green’s function and fixed point index theory, some sufficient conditions for the multiplicity and nonexistence of positive solutions to a class of nonlinear fractional boundary value problem are obtained. Three examples are given to show the effectiveness of our results.     

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.     

FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES

In this paper it has been systematically studied the imbedding properties o f fractional integral operatorsof periodic functions of several variables,and isomorphic properties of fractional intregral operators in thespaces of Lipschitz continuous functions. It has also been proved that the space of fractional integration,thespace of Lipschitz continuous functions and the Sobolev space are identical in L~2-norm.Results obtainedhereare not true for fractional integrals(or Riesz potentials)in R~n.     

Existence and Uniqueness of Positive Solutions for a System of Multi-order Fractional Differential Equations

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equa-tions. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.     

FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper,we introduce new concepts of α-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in [21,22] and different from α-GF-contractions given in [8].Then,sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings,in the setting of complete metric space.Consequently,the obtained results encompass various generalizations of the Banach contraction principle.Moreover,some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.     

FINITE DIFFERENCE FRACTIONAL STEP METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

Characteristic finite difference fractional step schemes are put forward. The electric potential equation is described by a seven-point finite difference scheme, and the electron and hole concentration equations are treated by a kind of characteristic finite difference fractional step methods. The temperature equation is described by a fractional step method. Thick and thin grids are made use of to form a complete set. Piecewise threefold quadratic interpolation, symmetrical extension, calculus of variations, commutativity of operator product, decomposition of high order difference operators and prior estimates are also made use of. Optimal order estimates in l2 norm are derived to determine the error of the approximate solution. The well-known problem is thorongley and completely solred.     

Fractional Wavelet Packet Transformations Involving Hankel–Clifford Integral Transformations

In this paper, we introduce the fractional wavelet transformations(FrWT) involving Hankel–Clifford integral transformation(HClIT) on the positive half line and studied some of its basic properties. Also we obtain Parseval's relation and an inversion formula. Examples of fractional powers of Hankel–Clifford integral transformation(FrHClIT) and FrWT are given. Then, we introduce the concept of fractional wavelet packet transformations FrBWPT and FrWPIT, and investigate their properties.     

OSCILLATIONS OF A CLASS OF HIGH-ORDER LINEAR ODE WITH IMPULSES

Oscillation of solutions for a class of nth-order linear differential equation with impulses are considered and some sufficient conditions for oscillation of solutions are obtained, which improve and popularize some results in parts of the relative references.     

Maximum fractional factors in graphs

We prove that fractional k-factors can be transformed among themselves by using a new adjusting operation repeatedly. We introduce, analogous to Berge’s augmenting path method in matching theory, the technique of increasing walk and derive a characterization of maximum fractional k-factors in graphs. As applications of this characterization, several results about connected fractional 1-factors are obtained.     

Graphs with the maximum or minimum number of 1-factors

Recently Alon and Friedland have shown that graphs which are the union of complete regular bipartite graphs have the maximum number of 1-factors over all graphs with the same degree sequence. We identify two families of graphs that have the maximum number of 1-factors over all graphs with the same number of vertices and edges: the almost regular graphs which are unions of complete regular bipartite graphs, and complete graphs with a matching removed. The first family is determined using the Alon and Friedland bound. For the second family, we show that a graph transformation which is known to increase network reliability also increases the number of 1-factors. In fact, more is true: this graph transformation increases the number of k-factors for all k≥1, and “in reverse” also shows that in general, threshold graphs have the fewest k-factors. We are then able to determine precisely which threshold graphs have the fewest 1-factors. We conjecture that the same graphs have the fewest k-factors for all k≥2 as well.     

PROPERTIES OF FRACTIONAL κ-FACTORS OF GRAPHS

In this paper the properties of some maximum fractional [0, κ]-factors of graphs are presented. And consequently some results on fractional matchings and fractional 1-factors are generalized and a characterization of fractional κ-factors is obtained.     

Toughness and the existence of fractional k-factors of graphs

The toughness of a graph G, t(G), is defined as t(G)=min{|S|/ω(G-S)|S⊆V(G),ω(G-S)>1} where ω(G-S) denotes the number of components of G-S or t(G)=+∞ if G is a complete graph. Much work has been contributed to the relations between toughness and the existence of factors of a graph. In this paper, we consider the relationship between the toughness and the existence of fractional k-factors. It is proved that a graph G has a fractional 1-factor if t(G)?1 and has a fractional k-factor if t(G)?k-1/k where k?2. Furthermore, we show that both results are best possible in some sense.     

Toughness, degrees and 2-factors

In this paper we generalize a Theorem of Jung which shows that 1-tough graphs with are hamiltonian. Our generalization shows that these graphs contain a wide variety of 2-factors. In fact, these graphs contain not only 2-factors having just one cycle (the hamiltonian case) but 2-factors with k cycles, for any k such that .     

Monochromatic geometric k-factors in red-blue sets with white and Steiner points

We study the existence of monochromatic planar geometric k-factors on sets of red and blue points. When it is not possible to find a k-factor we make use of auxiliary points: white points, whose position is given as a datum and which color is free; and Steiner points whose position and color is free. We present bounds on the number of white and/or Steiner points necessary and/or sufficient to draw a monochromatic planar geometric k-factor.     

The number of 1-factors in 2k-connected graphs

Mader [Arch. Math. 23 (1972), 219–224] determined the minimal number of 1-factors in a 2k-connected graph having at least one 1-factor. We give a simple proof of this result and slightly extend it.     

A bound for the number of vertices of a polytope with applications

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has {ie1-1} vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k≥2r+1 and every cut in the graph with at least two vertices on each side has more than k/r edges.     

Hamilton decompositions of graphs with primitive complements

A k-factor of a graph is a k-regular spanning subgraph. A Hamilton cycle is a connected 2-factor. A graph G is said to be primitive if it contains no k-factor with 1≤k<Δ(G). A Hamilton decomposition of a graph G is a partition of the edges of G into sets, each of which induces a Hamilton cycle. In this paper, by using the amalgamation technique, we find necessary and sufficient conditions for the existence of a 2x-regular graph G on n vertices which:

1.

has a Hamilton decomposition, and

2.

has a complement in K_n that is primitive.

This extends the conditions studied by Hoffman, Rodger, and Rosa [D.G. Hoffman, C.A. Rodger, A. Rosa, Maximal sets of 2-factors and Hamiltonian cycles, J. Combin. Theory Ser. B 57 (1) (1993) 69-76] who considered maximal sets of Hamilton cycles and 2-factors. It also sheds light on construction approaches to the Hamilton-Waterloo problem.