Perfect and Near-Perfect Matchings in Bilinear Congruence Graphs over Z_n

We study matching and covering parameters of the bilinear congruence graph 𝑮𝒏 over the ring ℤ𝒏. The graph has vertex set ℤ𝒏𝟐, and two distinct vertices (𝒂,𝒃) and (𝒙,𝒚) are adjacent whenever 𝒂𝒚≡𝒃𝒙 (𝐦𝐨𝐝 𝒏). This adjacency is a determinant-zero condition, so it describes modular dependence between ordered pairs rather than the usual product-zero relation from zero-divisor graphs. We prove that, for every odd integer 𝒏≥𝟑, the map (𝒂,𝒃)↦(−𝒂,−𝒃) gives a near-perfect matching of 𝑮𝒏. Consequently, 𝜈(𝐺𝑛)=𝑛2−12,𝜌(𝐺𝑛)=𝑛2+12.For an odd prime 𝒑, the proportionality class decomposition gives 𝜶(𝑮𝒑)=𝒑+𝟏,𝝉(𝑮𝒑)=𝒑𝟐−𝒑−𝟏.For distinct primes 𝒑 and 𝒒, we prove the exact independence formula 𝛼(𝐺𝑝𝑞)=(𝑝+1)(𝑞+1),and hence 𝜏(𝐺𝑝𝑞)=𝑝2𝑞2−(𝑝+1)(𝑞+1).In the case 𝒏=𝟐𝒒, where 𝒒 is an odd prime, we construct a perfect matching and deduce 𝜈(𝐺2𝑞)=𝜌(𝐺2𝑞)=2𝑞2.Finally, computations for 𝟐≤𝒏≤𝟑𝟎 support the study and suggest that 𝑮𝒏 may have a perfect matching for every even 𝒏≥𝟒.