Some results on ascending subgraph decomposition - Zhuo, Xin-Jian; Su, Yong-Mei (Utilitas Mathematica, 1999)
Let G be a graph of positive size q, and let n be that positive integer for which (n+12) ≤ q < (n+22). Then G is said to have an ascending subgraph decomposition if G can be decomposed into n subgraphs G1 , G2, ⋯ , Gn without isolated vertices such that Gi is isomorphic to a proper subgraph of Gi+1, for i = 1, 2, ⋯ , n -1. The graph G is shown to have an ascending subgraph decomposition when either G = Kn - Hn+[(n,-1)/2] (where Hn+[(n-1)/2] is a subgraph of Kn with at most n + [(n-1)/2] edges) or G is any complete bipartite graph.