An (F3,F5)-partition of planar graphs with girth at least 5

Abstract

Given a graph G=(V,E), if its vertex set V(G) can be partitioned into two non-empty subsets V1 and V2 such that Δ(G[V1])≤d1 and Δ(G[V2])≤d2, then we say that G admits a (Δd1,Δd2)-partition. If G[V1] and G[V2] are both forests with maximum degree at most d1 and d2, respectively, then we further say that G admits an (Fd1,Fd2)-partition.Let Gg denote the class of planar graphs with girth at least g. It is known that every graph in G5 admits a (Δ3,Δ5)-partition Choi and Raspaud (2015) [11]. In this paper, we strengthen this result by proving that every graph in G5 admits an (F3,F5)-partition.