The $ (a, d) $-ascending subgraph decomposition

Abstract

Let $ G $ be a graph of size $ q $ and $ a $, $ n $, $ d $ be positive integers for which $ \frac{n}{2}(2a+(n-1)d)\le q<(\frac{n+1}{2})(2a+nd) $. Then $ G $ is said to have $ (a, d) $- ascending subgraph decomposition into $ n $ parts $ ((a, d)- ASD) $ if the edge set of $ G $ can be partitioned into $ n $-non-empty sets generating subgraphs $ G_1 $, $ G_2 $, $ G_3 $, $ \ldots $, $ G_n $ without isolated vertices such that each $ G_i $ is isomorphic to a proper subgraph of $ G_{i+1} $ for $ 1\le i\le n-1 $ and $ |E(G_i)|=a+(i-1)d $. In this paper, we find $ (a, d)-ASD $ into $ n $ parts for $ W_m $.