On the existence of maximum resolvable -packings

Suppose K v is the complete undirected graph with v vertices and G is a finite simple undirected graph without isolated vertices. A G -packing of K v is a pair ( X , B ) , where X is the vertex set of K v and B is a collection of edge-disjoint subgraphs (blocks) isomorphic to G in K v . A G -packing ( X , B ) is called resolvable if B can be partitioned into parallel classes such that every vertex is contained in precisely one block of each class. Let ( v , G , 1 ) -MRP denote a resolvable G -packing containing the maximum possible number r ( v , G ) of parallel classes. Suppose e ( G ) and V ( G ) are the number of edges and the vertex set of the graph G , respectively. Let k = | V ( G ) | . Clearly, v ≡ 0 (mod k ) and r ( v , G ) ≤ ⌊ k ( v − 1 ) / 2 e ( G ) ⌋ for a ( v , G , 1 ) -MRP. Let K 4 − e be the graph obtained from a K 4 by removing one edge. It is proved in this paper that there exists a ( v , K 4 − e , 1 ) -MRP with ⌊ 2 ( v − 1 ) / 5 ⌋ parallel classes if and only if v ≡ 0 (mod 4) with the possible exceptions of v = 12 , 116 , 132 , 172 , 232 , 280 , 292 , 296 , 372 , 412 , 532 , 592 , 612 .