The Ascending Star Subgraph Decomposition of Some Star Forests

Let \(G\) be a graph of size \(\binom{n+1}{2}\) for some integer \(n \geq 2\). Then \(G\) is said to have an ascending star subgraph decomposition if \(G\) can be decomposed into \(n\) subgraphs \(G_1, G_2, \ldots, G_n\) such that each \(G_i\) is a star of size \(i\) with \(1 \leq i \leq n\). We shall prove in this paper that a star forest with size \(\binom{n+1}{2}\) possesses an ascending star subgraph decomposition under some conditions on the number of components or the size of components.