Let G be a graph of size ($\displaystyle\frac{n+1}{2}$) for some integer n ≥ 1. G is said to have an ascending subgraph decomposition (ASD) if can be decomposed into n subgraphs
H1, . . . ,Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1, i =
1, . . . , n−1. In this work we deal with ascending subgraph decompositions of bipartite graphs.
In order to do so, we consider ascending subgraph decompositions in which each
factor is a forest of stars. We show that every bipartite ...
Let G be a graph of size ($\displaystyle\frac{n+1}{2}$) for some integer n ≥ 1. G is said to have an ascending subgraph decomposition (ASD) if can be decomposed into n subgraphs
H1, . . . ,Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1, i =
1, . . . , n−1. In this work we deal with ascending subgraph decompositions of bipartite graphs.
In order to do so, we consider ascending subgraph decompositions in which each
factor is a forest of stars. We show that every bipartite graph G with($\displaystyle\frac{n+1}{2}$)edges
such that the degree sequence d1 ≥ · · · ≥ dk of one of the partite sets satisfies
d1 ≥ (k − 1)(n − k + 1), and di ≥ n − i + 2 for 2 ≤ i < k, admits an ASD with star forests. We also give a necessary condition on the degree sequence of G to have an ascending subgraph decomposition into star forests that is not far from the above
sufficient one. Our results are based on the existence of certain matrices that we
call ascending and the construction of edge-colorings of some bipartite graphs with
parallel edges.
Citation
Llado, A.; Moragas, J. On the ASD conjecture. A: VII Jornadas de Matemática Discreta y Algorítmica. "VII Jornadas de Matemática Discreta y Algorítmica". Castro Urdiales: 2010, p. 409-423.