Positive matching decompositions of the cartesian product of graphs

TL;DR

This work studies positive matching decompositions (pmd) under Cartesian products $\Gamma_1\square\Gamma_2$, establishing sharp upper bounds such as $\mathrm{pmd}(\Gamma_1\square\Gamma_2) \le \min\{ \mathrm{pmd}(\Gamma_1)\chi(\Gamma_2) + \mathrm{pmd}(\Gamma_2), \mathrm{pmd}(\Gamma_2)\chi(\Gamma_1) + \mathrm{pmd}(\Gamma_1) \}$ and developing a forest-decomposition/Latin-rectangle framework to obtain further bounds via induced subgraphs $F'_i$. It introduces the invariant $\kappa_{\Gamma}(n,p)$, linking $\mathrm{pmd}(\Gamma\square\Gamma')$ to acyclic edge-set covers and computes $\kappa_{\Gamma}(n,p)$ for cycles and other graph families, enabling precise pmd estimates for product graphs. The paper then applies these tools to grid graphs, deriving exact values such as $\mathrm{pmd}(P_m\square P_n)=4$ for $m,n\ge3$, and providing parity-based results for $P_m\square C_n$ and $C_m\square C_n$, including $\mathrm{pmd}(C_4\square C_4)=6$ and $\mathrm{pmd}(C_3\square C_3)=6$, with ancillary constructions like circular wall graphs to broaden the catalog of cases. Overall, the results connect combinatorial decompositions with algebraic perspectives on Lovász–Saks–Schrijver ideals and yield a detailed map of pmd behavior for the Cartesian product across key graph families.

Abstract

Let $Γ=(V,E)$ be a finite simple graph. A matching $M \subseteq E$ is positive if there exists a weight function on $V$ such that the matching $M$ is characterized by those edges with positive weights. A positive matching decomposition (pmd) of $Γ$ with $p$ parts is an ordered partition $E_1,\ldots,E_p$ of $E$ such that $E_i$ is a positive matching of $(V, E \setminus \bigcup_{j=1}^{i-1} E_j)$, for $i = 1, \ldots, p$. The smallest $p$ for which $Γ$ admits a pmd with $p$ parts is denoted by $\mathrm{pmd}(Γ)$. We study the pmd of the Cartesian product of graphs and give sharp upper bounds for them in terms of the pmds and chromatic numbers of their components. In special cases, we compute the pmd of grid graphs that is the Cartesian product of paths and cycles.

Figures (7)

• Figure 1: A non-prefix binary graph with $|A_0|=|A_{00}|=|A_{10}|=|A_{11}|=1$, $|A_{01}|=2$, and $|A_1|=3$
• Figure 2: Positive matchings $M_1$ (black) and $M_2$ (gray) of $P_m\square C_n$
• Figure 3: Positive matchings $M_1$ (black) and $M_2$ (gray) of $P_m\square C_n$
• Figure 4: Positive matching $M_1$
• Figure 5: Matchings $M_1$, $M_2$, and $M_3$ ($m$ odd and $n$ even)

Theorems & Definitions (60)

• Theorem 1.1: mfdg-sg-aayp
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Remark
• Definition
• Theorem 2.3
• proof
• Corollary 2.4