The proof of the Perepechko’s conjecture concerning near-perfect matchings on Cm x Pn cylinders of odd order

Rade Doroslovacki,Bojana Pantic,Olga Bodroza-Pantic,Jelena Djokic

doi:10.2298/aadm181016002d

Abstract

For all odd values of m, we prove that the sequence of the numbers of near-perfect matchings on Cm x P2n+1 cylinder with a vacancy on the boundary obeys the same recurrence relation as the sequence of the numbers of perfect matchings on Cm x P2n. Further more, we prove that for all odd values of m denominator of the generating function for the total number of the near-perfect matchings on Cm x P2n+1 graph is always the square of denominator of generating function for the sequence of the numbers of perfect matchings on Cm x P2n graph, as recently conjectured by Perepechko.

Highlights

The monomer-dimer problem, that of counting the exact number of coverings of a rectangular lattice by a previously specified number of monomers and dimers, arises in several models in statistical physics
Whereas in case the number of monomers is one the term we use for these configurations is near-perfect matchings
This paper deals with a monomer-dimer problem on Cm × Pn graphs which shall further on be referred to as cylinders (Figure 1)

Summary

INTRODUCTION

The monomer-dimer problem, that of counting the exact number of coverings of a rectangular lattice by a previously specified number of monomers and dimers, arises in several models in statistical physics. Let us denote by Dm ≡ (V (Dm), E(Dm)) (m ≥ 3) the digraph whose set of vertices V (Dm) consists of all the possible states of cycles Cm( j) of graph Gm,n for some perfect matching (n an arbitrary even integer) or near-perfect matching (n an arbitrary odd integer), whilst the set of edges E(Dm) is defined in the following way: there exists an edge from vertex p1 p2 . Each walk of length 2n in the digraph Dm which starts at a vertex from the set Bm∗ and ends at a vertex from the set Em defines a unique near-perfect matching of the graph Gm,2n+1 with a vacancy on the cycle Cm(1) whose state of the cycle Cm( j) for each j, where 1 ≤ j ≤ n, is exactly the word corresponding to the j-th vertex of that walk

Forming the above mentioned bijections we have that

MM L vL