A partition M is k-optimal if M minimizes BdM). For instance, if G has a Hamiltonian path J.Lo, then M = W.o} is a k -optimal partition. So the k -optimal partitions extend in some sense the concept of Hamiltonian paths. The theorem of Greene and Kleitman [10], which extends the Dilworth theorem [5], shows an important property of k-optimal partitions for the graph of a partially ordered set (i...

A graph G = (V,E) is arbitrarily partitionable (AP) if for any sequence τ = (n1, . . . , np) of positive integers adding up to the order of G, there is a sequence of mutually disjoints subsets of V whose sizes are given by τ and which induce connected graphs. If, additionally, for given k, it is possible to prescribe l = min{k, p} vertices belonging to the first l subsets of τ , G is said to be...

Let G be a graph of order n and 3 ≤ t ≤ n4 be an integer. Recently, Kaneko and Yoshimoto provided a sharp δ(G) condition such that for any set X of t vertices, G contains a hamiltonian cycle H so that the distance along H between any two vertices of X is at least n/2t. In this paper, minimum degree and connectivity conditions are determined such that for any graph G of sufficiently large order ...

Let G = (V,E) be a simple graph. A set D ⊆ V is a dominating set of G if every vertex of V − D is adjacent to a vertex of D. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. We prove that if G is a Hamiltonian graph of order n with minimum degree at least six, then γ(G) ≤ 6n 17 .

We propose a new discrete approximation to the Einstein equations, based on the Capovilla-Dell-Jacobson form of the action for the Ashtekar variables. This formulation is analogous to the Regge calculus in that the curvature has support on sets of measure zero. Both a Lagrangian and Hamiltonian formulation are proposed and we report partial results about the constraint algebra of the Hamiltonia...

We construct discrete analogues of Dirac structures by considering the geometry of symplectic maps and their associated generating functions, in a manner analogous to the construction of continuous Dirac structures in terms of the geometry of symplectic vector fields and their associated Hamiltonians. We demonstrate that this framework provides a means of deriving implicit discrete Lagrangian a...

Let G be a hamiltonian graph G of order n and maximum degree , and let C(G) denote the set of cycle lengths occurring in G. It is easy to see that |C(G)|¿ − 1. In this paper, we prove that if ¿n=2, then |C(G)|¿ (n + − 3)=2. We also show that for every ¿ 2 there is a graph G of order n¿ 2 such that |C(G)|= − 1, and the lower bound in case ¿n=2 is best possible. c © 2003 Elsevier Science B.V. All...

We interpret the Cayley transform of linear ( niteor in nitedimensional) state space systems as a numerical integration scheme of Crank Nicolson type. The scheme is known as Tustin's method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin's method is applied to a conservative (continuous time) linear system, then the resulting (discret...

We interpret the Cayley transform of linear (finiteor infinite-dimensional) state space systems as a numerical integration scheme of Crank–Nicolson type. The scheme is known as Tustin’s method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin’s method is applied to a conservative (continuous time) linear system, then the resulting (disc...

We present a new class of exponential integrators for ordinary differential equations. They are locally exact, i.e., they preserve the linearization of the original system at every point. Their construction consists in modifying existing numerical schemes in order to make them locally exact. The resulting schemes preserve all fixed points and are A-stable. The most promising results concern the...