Partitioning planar graph of girth 5 into two forests with maximum degree 4

Abstract

Given a graph G = (V, E), if we can partition the vertex set V into two nonempty subsets V₁ and V₂ which satisfy Δ(G[V₁]) ⩽ d₁ and Δ(G[V₂]) ⩽ d₂, then we say G has a (\({{\rm{\Delta }}_{{d_1}}}\,,{{\rm{\Delta }}_{{d_2}}}\))-partition. And we say G admits an (\({F_{d_{1}}}, {F_{d_{2}}}\))-partition if G[V₁] and G[V₂] are both forests whose maximum degree is at most d₁ and d₂, respectively. We show that every planar graph with girth at least 5 has an (F₄, F₄)-partition.