Self-orthogonal decompositions of graphs into matchings

Given a simple graph H, a self-orthogonal decomposition (SOD) of H is a collection of subgraphs of H, all isomorphic to some graph G, such that every edge of H occurs in exactly two of the subgraphs and any two of the subgraphs share exactly one edge. Our concept of SOD is a natural generalization of the well-studied orthogonal double covers (ODC) of complete graphs. If for some given G there is an appropriate H, then our goal is to find one with as few vertices as possible. Special attention is paid to the case when G a matching with n − 1 edges. We conjecture that v ( H ) = 2 n − 2 is best possible if n ≠ 4 is even and v ( H ) = 2 n if n is odd. We present a construction which proves this conjecture for all but 4 of the possible residue classes of n modulo 18.