On the vertex partition of planar graphs into forests with bounded degree

Abstract

Let G=(V,E) be a graph. A (Δd1,Δd2)-partition of a graph G is the partition of V(G) into two non-empty subsets V1 and V2, such that G[V1] and G[V2] are graphs with maximum degree at most d1 and d2, respectively. A similar definition can be given for the notation (Fd1,Fd2)-partition if G[V1] and G[V2] are forests with maximum degree at most d1 and d2, respectively.For any d1 and d2, Montassier and Ochem constructed graphs with girth 4 which do not admit (Δd1,Δd2)-partition. In this paper, we show that planar graphs with girth at least 5 and without adjacent 5-cycles admit (F3,F3)-partition.