Neighborhoods in Maximum Packings of 2 K n and Quadratic Leaves of Triple Systems

In this paper, two related problems are completely solved, extending two classic results by Colbourn and Rosa. In any partial triple system ( V , B ) of 2 K n , the neighborhood of a vertex v is the subgraph induced by { { x , y } ∣ { v , x , y } ∈ B } . For n ≡ 2 (mod 3) with n ≠ 2 , it is shown that for any 2‐factor F on n − 1 or n − 2 vertices, there exists a maximum packing of 2 K nwith triples such that F is the neighborhood of some vertex if and only if( n , F ) ≠ ( 5 , C 2 ∪ C 2 ) , thus extending the corresponding result for the case where n ≡ 0 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 λ ≥ 2 , thereby extending the solution where λ = 1 by Colbourn and Rosa.