On ascending subgraph decomposition of graphs

Abstract

Let G be a graph of positive size q, and let n be that positive integer for which . It was conjectured by Alavi et al that the G can be decomposed into subgraphs G₁, G₂, … ,G_n such that G_i is isomorphic to a proper subgraph of G_i+1 for 1 ≤ i ≤ n−1. Since the conjecture was posed around 25 years many special classes of graphs have been verified to have ascending subgraph decomposition defined above. Because of the interesting nature of the structure of graphs, many researchers have devoted themselves to work on this problem and also provided several extended versions of such decompositions. Mainly, the size of G and the structure of G_i’s are concerned. As hard as we work on this problem, we find the fascinating inside of this seemingly easy conjecture. This is the reason, we decide to make a survey of this decomposition and hopefully, we would (be very happy to) see the ASD conjecture proved to be true in the near future by some talented researchers.

Keywords:

• ascending subgraph decomposition of graph
• subset partition
• star
• path
• matching

Mathematics Subject Classification:

• 05C99