Decomposition of sparse graphs into two forests, one having bounded maximum degree
Let G be a graph. The maximum average degree of G, written Mad(G), is the largest average degree among the subgraphs of G. It was proved in Montassier et al. (2010) [11] that, for any integer k >= 0, every simple graph with maximum average degree less than m(k) = 4(k+1)(k+3)/k(2)+6k+6 admits an edge-partition into a forest and a subgraph with maximum degree at most k; furthermore, when k <= 3 both subgraphs can be required to be forests. In this note, we extend this result proving that, for k = 4, 5, every simple graph with maximum average degree less than m(k) admits an edge-partition into two forests, one having maximum degree at most k (i.e. every graph with maximum average degree less than 70/23 (resp. 192/61) admits an edge-partition into two forests, one having maximum degree at most 4 (resp. 5)). (C) 2010 Elsevier B.V. All rights reserved.
decomposition combinatorial problems forest with bounded degree discharging procedure global rules