Decomposition of sparse graphs into two forests, one having bounded maximum degree

Abstract

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 + 6 k + 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)).