A note on ascending subgraph decompositions of complete multipartite graphs (Q1841928)

The ascending subgraph decomposition conjecture is the following. Let \(G\) be a graph of size \(({n+1\over 2})\leq|E(G)|< ({n+2\over 2})\). Then \(E(G)\) can be partitioned into \(n\) sets \(E_1,E_2,\dots, E_n\) with induced subgraphs \(G_1,G_2,\dots, G_n\) such that \(|E(G_i)|<|E(G_{i+1})|\) and \(G_i\) is isomorphic to a subgraph of \(G_{i+1}\) for \(i= 1,2,\dots, n\). The authors show that the ascending subgraph decomposition conjecture is true for complete multipartite graphs.