Lower bound for the pmd of Cartesian products

Establish a universal constant c>0 such that pmd(Γ1)+pmd(Γ2)≤pmd(Γ1□Γ2)+c for all graphs Γ1 and Γ2.

Background

The results in Section 2 primarily provide upper bounds for the pmd of Cartesian products, while the authors identify only the maximum valency as a general lower bound. They conjecture that the pmd of a Cartesian product is, up to a universal additive constant, at least the sum of the pmds of its two factors.

References

However, we believe that the pmd of the Cartesian product of two graphs, scaled up to a constant value, is bounded below by sum of the pmd of its components.Conjecture 2.10. There exists a constant c > 0 such thatpmd(Γ1) + pmd(Γ2) ≤ pmd(Γ1□Γ2) + cfor all graphs Γ1 and Γ2.

— Positive matching decompositions of the cartesian product of graphs  (2502.02826 - Ghouchan et al., 5 Feb 2025) in Conjecture 2.10, Section 2, p. 7