On the ascending subgraph decomposition problem for bipartite graphs

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with ( n+1 2 ) edges admits an edge decomposition G = H1 ⊕ · · · ⊕Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1, i = 1, . . . , n−1. We show that every bipartite graph G with ( n+1 2 ) edges such that the degree sequence d1, . . . , dk of one of the stable sets satisfies di ≥ n − i + 2, 1 ≤ i < k, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.