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On the ASD conjecture
Lladó Sánchez, Ana M.; Moragas Vilarnau, Jordi
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV; Universitat Politècnica de Catalunya. COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions
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.
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
Graph theory
Matrices
Grafs, Teoria de
Matrius (Matemàtica)
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