Positive matching decomposition number of path-cycle grids

Prove that for all m,n≥3, pmd(Pm□Cn)=5 whenever n is even and n<4(m−1) or n is odd and n<2m.

Background

Theorem 4.1 establishes the general bounds 4≤pmd(Pm□Cn)≤5 and proves pmd(Pm□Cn)=4 for sufficiently large n, with the threshold depending on the parity of n.

Direct calculations for several small instances, including P3□C3 through P3□C6, yield pmd equal to 5. These computations motivate the conjecture that the remaining parameter ranges—short even cycles and short odd cycles relative to the path length—always have pmd exactly 5.