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

Abstract

We study matching and covering parameters of the bilinear congruence graph  over the ring . The graph has vertex set , and two distinct vertices  and  are adjacent whenever . 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 , the map  gives a near-perfect matching of .