On the ascending subgraph decomposition problem for bipartite graphs

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with View the MathML source(n+12) edges admits an edge decomposition G=H1¿¿¿HnG=H1¿¿¿Hn such that HiHi has i edges and is isomorphic to a subgraph of Hi+1Hi+1, i=1,…,n-1i=1,…,n-1. We show that every bipartite graph G with View the MathML source(n+12) edges such that the degree sequence d1,…,dkd1,…,dk of one of the stable sets satisfies di=n-i+2di=n-i+2, 1=i<k1=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.Peer Reviewe