A note on the ascending subgraph decomposition problem

Let G be a graph with (” : ‘) edges. We say G has an ascending subgraph decomposition (ASD) if the edge set of G can be partitioned into n sets generating graphs G,, G,, . , G, such that IE(G,)I = i (for i = 1, 2, . . , n) and G, is isomorphic to a subgraph of G,+r for i = 1,2, . . , n 1. In this note, we prove that if G is a graph of maximum degree d C [(n + 1)/2j on (” l ‘) edges, then G has an ASD. Moreover, we show that if d s [(n 1)/2], then G has an ASD with each member a matching. Subsequently, we also verify that every regular graph of degree a prime power has an ASD.