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

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.