Near-Perfect Matchings on Cylinders Cm × Pn of Odd Order

A close relationship was established between the number of perfect and nearperfect matchings on cylinders Cm×Pn. Generating functions are obtained for the number of near-perfect matchings in these graphs for fixed odd m 13. A conjecture is put forth on the properties of the denominators of generating functions for arbitrary odd m.


Introduction
One of the classical lattice models of statistical mechanics is the dimer problem. In this model, the rigid dimers (diatomic molecules) are treated as the edges of a graph together with the vertices incident to it. Each lattice node cannot be occupied by more than one atom, so the edges must be non-adjacent.
In the dimer model, close-packed lattice coverings containing the maximum possible number of dimers are studied. The combinatorial interpretation of the model leads to the need to enumerate the maximum matchings in certain set of lattice graphs.
For many kinds of lattices of even order, used in applications, the maximum matchings are perfect matchings. Near-perfect matchings are a natural analogue of perfect matchings in graphs of odd order. A near-perfect matching is one in which exactly one vertex is unmatched. A vertex of the graph, which is not saturated by a matching, will be called vacancy.
A significant part of the work related to the enumeration of matchings deals with perfect matchings. At the same time, the quantitative characteristics of other kinds of matchings require further study. In this paper cylinders C m × P n are considered when both parameters m and n are odd. Among the works of the past years devoted to the study of such graphs, the paper [1] is of the greatest interest. This paper gives a closed-form expression for the number of near-perfect matchings when the vacancy is on the boundary of a cylinder. In [2] Kong briefly mentions the results of numerical calculations on cylinders for a fixed value of the parameter m. The author reports about the similarity of asymptotic expansions for the total number of near-perfect matching on cylinders and rectangular lattices P m ×P n , while omitting the details of the computations.
In our work we use a universal method for enumerating matchings. This allows us to avoid many of the restrictions inherent to the Pfaffian technique, and to study in detail the dependence of the number of near-perfect matchings on the vacancy location. The results of investigation clearly demonstrate that parity considerations are just as important for solving this problem as for counting perfect matchings.
Extending the results of numerical calculations by Kong, we reveal a deep connection between the number of near-perfect and perfect matchings on the cylinder with the same value of m. In fact, for e-mail: persn@newmail.ru fixed m and changing n, the number of near-perfect matchings and the number of perfect matchings are solutions of the same recurrence relation if the vacancy location does not change. Based on this result, we can conjecture the form of the denominator of the generating function for the total number of near-perfect matchings on the cylinder for any odd m.

Basic notation
Consider the graph G m,n = C m × P n . This graph can be viewed as n copies of C m placed sequentially with edges joining the corresponding vertices of C m . We number all the cycles C m by integers from 1 to n. The ordinal number put in parentheses will be indicated as superscript as shown at the left of Figure 1. The union of graphs ∪ n k=1 C (k) m forms a spanning subgraph of G m,n . Let V be the set of vertices of G m,n and K v m (n), (v ∈ V) -the number of near-perfect matchings on the cylinder when the vacancy is fixed at the node v. We denote by V k the set of vertices of the cycle  The exponential dependence of the number of near-perfect matchings on both parameters of the graph makes it difficult to compare different profiles. It is convenient to introduce normalized profiles by dividing allK (k) m (n) byK (1) m (n). An example of such a profile for a small cylinder is shown at the right in Figure 1. The main advantage of the normalized profiles is their weak dependence on the parameters m and n.
For the existence of a near-perfect matching in the graph, its order must be odd, so in the future the parameter m will always be considered odd. The different kinds of maximum matchings discussed in this paper will be distinguished by superscripts as follows:

Counting matchings
The method proposed by Wilf was used to count matchings [3]. This method allows us to reduce the problem of counting perfect matchings in a graph of order n to evaluating the coefficients of a certain polynomial in n variables. It is attractive due to its universality and ease of implementation in computer algebra systems. In our implementation of the algorithm we used a 64-bit version of Maple 17. Due to the extensive set of library functions built into this CAS, the program code turned out to be very compact.
To generate the graphs studied, the GraphTheory package was used. The same package was used to perform all operations on graphs. Calling routine CartesianProduct(PathGraph(n), CycleGraph(m)) with odd m and n created a cylinder of odd order. To calculate the profile of near-perfect matchings, one vertex belonging to the cycle C (k) m was selected for each value of k from 1 to n/2 . This vertex, together with the edges incident to it, was removed from the cylinder. Then, to evaluateK (k) m (n), the algorithm by Wilf was applied to the resulting even-order graph. The second half of the profile was reconstructed on the basis of symmetry considerations. To accelerate the computations, the values ofK (k) m (n) for different k were calculated in parallel. To do this, the tools of the Threads package were used.
After calculating the profile of near-perfect matchings on the graph G m,n , we can find the value of K N m ( n/2 ), which was of primary interest in the paper by Kong. Then the parameter n was increased by 2 and all calculations were repeated on the graph G m,n+2 . Multiple repetition of the calculations made it possible to obtain initial segments of the sequences {K (k) m (n)} and {K N m (n)}. In those cases when these segments turned out to be sufficiently long, it was possible to derive recurrence relations. For example, the sequence {K B 3 (n)} satisfies a second-order relationK B 3 (n + 1) = 5K B 3 (n) −K B 3 (n − 1). However, {K P 3 (n)} also satisfies the same recurrence relation, which indicates close connections between perfect and near-perfect matchings.
For Cartesian product graphs and fixed values of m, the method by Wilf has properties which are typical for algorithms from the complexity class FPT. The resources required to computeK (k) m (n) depend exponentially on m and are polynomial in n. In turn, the upper bound of the order of the recurrence relations satisfied by the sequence {K P m (n)} is equal to 2 m/2 , and this boundary as a rule is achieved [4]. To derive such relation, the initial segment must contain more than 2 m/2 +1 elements. Under such circumstances, obtaining analytical expressions is possible only for small m. At moderate values of m, we must limit ourselves to the numerical values of the number of near-perfect matchings.
All calculations were carried out on a personal computer with an Intel Core i7-980 3.33 GHz processor and 24 GB RAM. The computer was running a 64-bit version of Windows 7. Calculations were performed in the 64-bit version of Maple 17. Available facilities make it possible to find linear recurrence relations and generating functions G B m (z), G N m (z) on cylinders for m 13. Profiles of nearperfect matchings were evaluated on G m,n graphs for 15 m 19 and n n * , where n * = 135 for m = 15, 17, and n * = 85 for m = 19.

Main results
In this section, we will discuss, on simple examples, those properties of near-perfect matchings that hold for all the studied values of m. The generating functions G P m (z) will be considered known, since they were found in [4] for m 30.