The H-Decomposition Problem for Graphs

The concept of H-decompositions of graphs was first introduced by Erdös, Goodman and Pósa in 1966, who were motivated by the problem of representing graphs by set intersections. Given graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let be the smallest number  , n H    , such that, any graph of order n admits an H-decomposition with at most  parts. The exact computation of   , n H  for an arbitrary H is still an open problem. Recently, a few papers have been published about this problem. In this survey we will bring together all the results about H-decompositions. We will also introduce two new related problems, namely Weighted H-Decompositions of graphs and Monochromatic H-Decompositions of graphs.