Exact pmd of path-cycle grid graphs in the remaining parameter range

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

Background

Proposition 4.1 proves that 4≤pmd(Pm□Cn)≤5 for all m,n≥3 and establishes pmd(Pm□Cn)=4 in the complementary large-n regimes: even n≥4(m−1) and odd n>2m. Direct calculations for several small instances yield value 5, motivating the conjecture that the upper bound is sharp throughout the remaining range.

References

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

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