Abstract: Alavi and his fellows defined the concept of ascending subgraph decomposition of a graph and conjectured that every graph with positive size has an ascending subgraph decomposition in paper [1]. Paper [2] proved that K_n-R_n-1 has a star ascending subgraph decomposition,here K_n is the complete graph with order n and R_n-1 is a subgraph of K n with size at most n-1. In paper [3],Ma Kejie and Chen Huaitang proved that K_n-R_n has an ascending subgraph decomposition when the size of R n is not greater than n. In this paper we will prove K_n-R has an ascending subgraph decomposition when the size of R is less than 3n/2. This paper will also give the concept of comet and prove that K_n-R_n-1 has a comet ascending subgraph decomposition.  

Key words:  , graph；comet；ascending subgraph decomposition；conjecture