• neighborhood graphs;
  • graph decompositions;
  • quadratic leaves;
  • partial triple systems


In this paper, two related problems are completely solved, extending two classic results by Colbourn and Rosa. In any partial triple system inline image of inline image, the neighborhood of a vertex v is the subgraph induced by inline image. For inline image (mod 3) with inline image, it is shown that for any 2-factor F on inline image or inline image vertices, there exists a maximum packing of inline image with triples such that F is the neighborhood of some vertex if and only if inline image, thus extending the corresponding result for the case where inline image or 1 (mod 3) by Colbourn and Rosa. This result, along with the companion result of Colbourn and Rosa, leads to a complete characterization of quadratic leaves of λ-fold partial triple systems for all inline image, thereby extending the solution where inline image by Colbourn and Rosa.