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On the number of Hamiltonian cycles of P4 x Pn
... Studies on the topic of counting Hamiltonian cycles in different families of graphs and the use of similar (matrix) approaches are not negligible. In 1990, a characterization of Hamiltonian cycles of the Cartesian product P 4 × P n was established [10]. In 1994, Kwong and Rogers developed a matrix method for counting Hamiltonian cycles in P m × P n , obtaining exact results for m = 4, 5 [11]. ...
... ,c r represents the number of possible combinations in T i for k distinct paths covering all internal vertices that have ordered left wall endvertices b p and ordered right wall endvertices c r (defining paths b 1 1-traversing: In this case, the sets L e , R e , L r , R r are the following: 10, 01, 00}. ...
Recently, the problem of counting Hamiltonian cycles in 2-tiled graphs was resolved by Vegi Kalamar, Bokal, and Žerak. In this paper, we continue our research on generalized tiled graphs. We extend algorithms on counting traversing Hamiltonian cycles from 2-tiled graphs to generalized tiled graphs. We further show that, similarly as for 2-tiled graphs, for a fixed finite set of tiles, counting traversing Hamiltonian cycles can be performed in linear time with respect to the size of such graph, implying that counting traversing Hamiltonian cycles in tiled graphs is fixed-parameter tractable.
... Studies on the topic of counting Hamiltonian cycles in different families of graphs and the use of similar (matrix) approaches are not negligible. In 1990, a characterization of Hamiltonian cycles of the Cartesian product P 4 P n was established [7]. In 1994, Kwong and Rogers developed a matrix method for counting Hamiltonian cycles in P m P n , obtaining exact results for m = 4, 5 [8]. ...
In this paper we extend counting of traversing Hamiltonian cycles from 2-tiled graphs to generalized tiled graphs. We further show that, for a fixed finite set of tiles, counting traversing Hamiltonian cycles can be done in linear time with respect to the size of such graph, implying counting Hamiltonian cycles in tiled graphs is fixed-parameter tractable.
... A closed Hamiltonian path is called a Hamiltonian cycle or Hamiltonian circuit, which we shall abbreviate as HC. The enumeration of Hamiltonian cycles on rectangular grid graphs P m × P n had been studied extensively in, among others, [2,4,9,15,10,13,14,17,19,20]. In contrast, little work [2,9,11,17] was devoted to enumerate Hamiltonian cycles on rectangular grid cylinders C m × P n . ...
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.
... This shows an interesting intuitive duality to counting Eulerian cycles that showed relevance in constructing controlled, de novo protein structure folding [10,11]. In 1990, a characterization of Hamiltonian cycles of the Cartesian product P 4 P n was established [12]. In 1994, Kwong and Rogers developed a matrix method for counting Hamiltonian cycles in P m P n , obtaining exact results for m = 4, 5 [13]. ...
In 1930, Kuratowski showed that K3,3 and K5 are the only two minor-minimal nonplanar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň and Kochol showed that there are infinitely many k-crossing-critical graphs for any k≥2, even if restricted to simple 3-connected graphs. Recently, 2-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodroža-Pantić, Kwong, Doroslovački, and Pantić.
... This shows an interesting intuitive duality to counting Eulerian cycles that showed relevance in constructing controlled, de novo protein structure folding [2,21]. In 1990, a characterization of Hamiltonian cycles of the Cartesian product P 4 P n was established [35]. In 1994, Kwong and Rogers developed a matrix method for counting Hamiltonian cycles in P m P n , obtaining exact results for m = 4, 5 [27]. ...
In 1930, Kuratowski showed that and are the only two minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. \v{S}ir\'{a}\v{n} and Kochol showed that there are infinitely many k-crossing-critical graphs for any , even if restricted to simple 3-connected graphs. Recently, 2-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodro\v{z}a-Panti\'c, Kwong, Doroslova\v{c}ki, and Panti\'c for .
... Many efforts have been devoted to the enumeration of Hamiltonian cycles and related problems in a rectangular grid graph P m × P n+1 . They are documented in, among others, [1,4,7,8,10,13,14,15,19,20]. The transfer matrix method [5,18] provides a powerful tool in this regard. ...
19 We continue our research in the enumeration of Hamiltonian cycles (HC) 20 on thin cylinder grid graphs C m × P n+1 by studying a triangular variant of 21 the problem. There are two types of HCs, distinguished by whether they 22 wrap around the cylinder. Using two characterizations of these HCs, we 23 prove that, for fixed m, the number of HCs of both types satisfy some linear 24 recurrence relations. For small m, computational results reveal that the two 25 numbers are asymptotically the same. We conjecture that this is true for all 26 m ≥ 2.
... Enumeration of Hamiltonian cycles and related problems in rectangular grids had been extensively studied. See, for examples, [2,[4][5][6][11][12][13][14][17][18][19]25,26]. A common thread among these papers is the introduction of an encoding method, and the use of a transfer matrix (with the exception in [2,17]) to study the transition between the underlying structures within the Hamiltonian cycles. ...
In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate the thermodynamics of protein folding. In this paper, we present new characterizations of the Hamiltonian cycles in labeled triangular grid graphs, which are graphs constructed from rectangular grids by adding a diagonal to each cell. By using these characterizations and implementing the computational method outlined here, we confirm the existing data, and obtain some new results that have not been published. A new interpretation of Catalan numbers is also included.
... A closed Hamiltonian path is called a Hamiltonian cycle or Hamiltonian circuit, which we shall abbreviate as HC. The enumeration of Hamiltonian cycles on rectangular grid graphs P m × P n had been studied extensively in, among others, [2,4,9,15,10,13,14,17,19,20]. In contrast, little work [2,9,11,17] was devoted to enumerate Hamiltonian cycles on rectangular grid cylinders C m × P n . ...
We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph Cm x Pn+1. We distinguish two types of Hamiltonian cycles depending on their contractibility (as Jordan curves) and denote their numbers hmnc (n) and hmc (n). For fixed m, both of them satisfy linear homogeneous recurrence relations with constant coefficients. We derive their generating functions and other related results for m ≤ 10. The computational data we gathered suggests that hmnc (n) ∼ hmc (n) when m is even. © 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS).
... Mathematicians have dealt with this counting problem for two-dimensional lattices, i.e. m × n grid graph (m, n ∈ N ), too. The enumeration of Hamiltonian cycles, abbreviated HC, for small values of m, fixing m and letting n grow, was studied by ad hoc methods [9,16,17,22]. An algorithm which allows us to systematically compute generating functions for these sequences (with counter n) for any m was for the first time described in [1] (less known) and somewhat later, independently in [21]. ...
In polymer science, Hamiltonian paths and Hamiltonian circuits can serve as excellent simple models for dense packed globular proteins. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate thermodynamics of protein folding. Hamiltonian circuits are a mathematical idealization of polymer melts, too. The number of Hamiltonian cycles on a graph corresponds to the entropy of a polymer system. We present new characterizations of the Hamiltonian cycles in a labeled rectangular grid graph P m ×P n and in a labeled thin cylinder grid graph C m ×P n . We proved that for any fixed m, the numbers of Hamiltonian cycles in these grid graphs, as sequences with counter n, are determined by linear recurrences. The computational method outlined here for finding these difference equations together with the initial terms of the sequences has been implemented. The generating functions of the sequences are given explicitly for some values of m. The obtained data are consistent with data obtained in the works by Kloczkowski and Jernigan, and Schmalz et al.
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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