THE PROOF OF THE PEREPECHKO’s CONJECTURE CONCERNING NEAR-PERFECT MATCHINGS ON Cm × Pn CYLINDERS OF ODD ORDER

For all odd values of m, we prove that the sequence of the numbers of near-perfect matchings on Cm × P 2n+1 cylinder with a vacancy on the boundary obeys the same recurrence relation as the sequence of the numbers of perfect matchings on Cm × P 2n . 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 × P 2n+1 graph is always the square of denominator of generating function for the sequence of the numbers of perfect matchings on Cm × P 2n graph, as recently conjectured by Perepechko.