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The structure of the 2-factor transfer digraph common for thin cylinder, torus and Klein bottle grid graphs
Abstract
We prove that the transfer digraph D* C,m needed for the enumeration of 2-factors in the thin cylinder TnCm(n), torus TGm(n) and Klein bottle KBm(n) (all grid graphs of the fixed width m and with m?n vertices), when m is odd, has only two components of order 2m?1 which are isomorphic. When m is even, D* C,m has [m/2] + 1 components which orders can be expressed via binomial coefficients and all but one of the components are bipartite digraphs. The proof is based on the application of recently obtained results concerning the related transfer digraph for linear grid graphs (rectangular, thick cylinder and Moebius strip).
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In this paper, we propose an algorithm for obtaining the common transfer digraph Dm* for enumeration of 2-factors in graphs from the title, all of which have m·n vertices (m,n∈N,m≥2). The numerical data gathered for m≤18 reveal some matches for the numbers of 2-factors for different types of torus or Klein bottle. In the latter case, we conjecture that these numbers are invariant under twisting.
In this paper, we prove that all but one of the components of the transfer digraph D? m needed for the enumeration of 2-factors in the rectangular, thick cylinder and Moebius strip grid graphs of the fixed width m (m ? N) are bipartite digraphs and that their orders could be expressed in term of binomial coefficients. In addition, we prove that the set of vertices of each component consists of all the binary m-words for which the difference of numbers of zeros in odd and even positions is constant.
Here, in Part II, we proceeded further with the enumeration of Hamiltonian cycles (HC's) on the grid cylinder graphs of the form Pm+1?Cn, where n is allowed to grow and m is fixed. We proposed two novel characterisations of the contractible HC's. Finally, we made a conjecture concerning the dependency of the asymptotically dominant type of HC's on the parity of m.
Graph Theory
International audience
We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph . We distinguish two types of Hamiltonian cycles, and denote their numbers and . For fixed m, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for . The computational data we gathered suggests that when m is even.
In a recent paper, we have studied the enumeration of Hamiltonian cycles (abbreviated HCs) on the grid cylinder graph P m+1 × C n , where m grows while n is fixed. In this sequel, we study a much harder problem of enumerating HCs on the same graph only this time letting n grow while m is fixed. We propose a characterization for non-contractible HCs which enables us to prove that their numbers h mnc (n) satisfy a recurrence relation for every fixed m. From the computational data, we conjecture that the coefficient for the dominant positive characteristic root in the explicit formula for h mnc (n) is 1.
We present an algorithm for enumerating exactly the number of Hamiltonian chains on regular lattices in low dimensions. By definition, these are sets of k disjoint paths whose union visits each lattice vertex exactly once. The well-known Hamiltonian circuits and walks appear as the special cases k=0 and k=1 respectively. In two dimensions, we enumerate chains on L x L square lattices up to L=12, walks up to L=17, and circuits up to L=20. Some results for three dimensions are also given. Using our data we extract several quantities of physical interest.
The result of a collaboration between a theoretician and an experimentalist, this book is devoted to the static properties of flexible polymers in solution. It presents the vast progress made by both theory and experiment in recent years. Despite the variety in the chemical composition and physical properties of long polymer chains, when in solution they show a universality in their behaviour. On the experimental side, the use of photon and neutron scattering has led to a better understanding, while the use of computer simulation has also produced interesting results. This work is the result of a collaboration between a theoretician and an experimentalist, who have both worked for many years on polymer solutions.
As digital microfluidic biochips (DMFBs) make the transition to the marketplace for commercial exploitation, security and intellectual property (IP) protection are emerging as important design considerations. Recent studies have shown that DMFBs are vulnerable to reverse engineering aimed at stealing biomolecular protocols (IP theft). The IP piracy of proprietary protocols may lead to significant losses for pharmaceutical and biotech companies. The microelectrode dot array (MEDA) is a next-generation DMFB platform that supports real-time sensing of droplets and has the added advantage of important security protection. However, real-time sensing offers opportunities to an attacker to steal the biochemical IP. We show that the daisychaining of microelectrodes and the use of one-time programmability in MEDA biochips provides effective bitstream scrambling of biochemical protocols. To examine the strength of this solution, we develop a Satisfiability (SAT)-based attack that can unscramble the bitstreams through repeated observations of bioassays executed on the MEDA platform. Based on insights gained from the SAT attack, we propose an advanced defense against IP theft. Simulation results using real-life biomolecular protocols confirm that while the SAT attack is effective for simple instances, our advanced defense can thwart it for realistic MEDA biochips and real-life protocols.
Given a pair of 1-complex Hamiltonian cycles C and in an L-shaped grid graph G, we show that one is reachable from the other under two operations, flip and transpose, while remaining in the family of 1-complex Hamiltonian cycles throughout the reconfiguration. Operations flip and transpose are local in G. We give a reconfiguration algorithm that uses O(|G|) operations.
The transfer matrix method has been developed to enumerate and generate compact self-avoiding walks in two dimensions on the square lattice within rectangular strips of size m×n. The method is significantly superior to the traditional method of computer generation of self-avoiding walks, because it is attrition-free, i.e., each computation leads to successful conformations, with no failures. The method is generalized to irregular shapes, and the extension of the method to the Monte Carlo sampling of the compact conformational space is proposed. Application of this new method to protein conformation generation is discussed, with the possibility of including several types of constraints. © 1998 American Institute of Physics.
Close-packed self-avoiding walks and circuits, as models for condensed polymer phases, are studied on the square-planar and honeycomb lattices. Exact solutions for strips from these lattices are obtained via transfer matrix methods. Extrapolations are made for the leading asymptotic terms in the count of compact conformations on the square-planar lattice. The leading asymptotic term for each lattice is bounded from below, and it is noted that boundary effects can be important.
A spanning union of cycles in rectangular grid graphs, thick grid cylinders and Moebius strips
- J Đokić
- O Bodroža-Pantić
- K Doroslovački