Positive matching decomposition number of cylindrical grid graphs

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

Background

Theorem 4.1 proves that 4≤pmd(Pm□Cn)≤5 and establishes pmd(Pm□Cn)=4 for sufficiently large n, with thresholds depending on the parity of n. Direct calculations give value 5 for several small instances.

The conjecture asserts that the remaining parameter ranges below those thresholds always have pmd equal to 5.

References

This motivates us to pose the following: Conjecture 4.2. If m, n ≥ 3, then pmd(Pm□Cn) = 5 whenever n is even and n < 4(m − 1) or n is odd and n < 2m.

— Positive matching decompositions of the cartesian product of graphs  (2502.02826 - Ghouchan et al., 5 Feb 2025) in Conjecture 4.2, Section 4, p. 15