Definition 1 A Hamiltonian cycle in a graph is a cycle that visits each vertex exactly once. A Hamiltonian graph is a graph that contains a Hamiltonian cycle. It is well known that the problem of determining if a graph is Hamiltonian is N P-complete. Here we will construct a NIZK proof in the hidden-bits model (HBM) that is able to prove that a graph is Hamiltonian. First we define how graphs a...

Let (M,ω) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with ω. For instance, g could be Kähler, with Kähler form ω. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional Volg under Hamiltonian deformations, computing Volg (L) using g|L. It is called Hamiltonian stable ...

Hamiltonian Monte Carlo and slice sampling are amongst the most widely used and studied classes of Markov Chain Monte Carlo samplers. We connect these two methods and present Hamiltonian slice sampling, which allows slice sampling to be carried out along Hamiltonian trajectories, or transformations thereof. Hamiltonian slice sampling clarifies a class of model priors that induce closed-form sli...

Z-mapping graph is a balanced bipartite graph G of a digraph D by split each vertex of D into a pair of vertices of G. Based on the property of the G, it is proved that if D is strong connected and G is Hamiltonian, then D is Hamiltonian. It is also proved if D is Hamiltonian, then G contains at least a perfect matching. Thus some existence sufficient conditions for Hamiltonian digraph and Hami...

We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase ...

The perturbation theory for purely imaginary eigenvalues of Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations is discussed. It is shown that there is a substantial difference in the behavior under these perturbations. The perturbation of real eigenvalues of real skew-Hamiltonian matrices under structured perturbations is discussed as well and these results are used to ana...

The prism over a graph G is the Cartesian productG K2 of G with the complete graph K2. If the prism over G is hamiltonian, we say that G is prism-hamiltonian. We prove that triangulations of the plane, projective plane, torus, and Klein bottle are prism-hamiltonian. We additionally show that every 4-connected triangulation of a surface with sufficiently large representativity is prism-hamiltoni...

A graph G is called hamiltonian-<:onnectedfrom a vertex v ifa hamiltonian path exists from v to every other vertex w ~ v. We present a survey of the main results known about such graphs, including a section on graphs uniquely hamiltonian-connected from a venex and a section on the computational complexity ofdetermining whether a given graph is hamiltonian-<:onnectedfrom a venex or uniquely hami...

A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can b...

When dissipative terms are dropped, all of the important models of plasma physics are described by partial differential equations that possess Hamiltonian form in terms of noncanonical Poisson brackets. For example, this is the case for ideal magnetohydrodynamics, the Vlasov–Maxwell equations, and other systems see Refs. 7–9 for review . Among these, there exist several reduced fluid models who...