On the vertex partitions of sparse graphs into an independent vertex set and a forest with bounded maximum degree

Given a graph G = (V, E), if its vertex set V (G) can be partitioned into two non-empty subsets V-1 and V-2 such that G [V-1] is edgeless and G [V-2] is a graph with maximum degree at most k, then we say that G admits an (I, Delta(k))-partition. A similar definition can be given for the notation (I, F-k)-partition if G [V-2] is a forest with maximum degree at most k. The maximum average degree of G is defined to be mad (G) = max {2 vertical bar E(H)vertical bar/vertical bar V(H)vertical bar : H subset of G}. Borodin and Kostochka (2014) proved that every graph G with mad (G) <= 8/3 admits an (I, Delta(2))-partition and every graph G with mad (G) <= 14/5 admits an (I, Delta(4))-partition. In this paper, we obtain a strengthening result by showing that for any k >= 2, every graph G with mad (G) <= 2 + k/k+1 admits an (I, F-k)-partition. As a corollary, every planar graph with girth at least 7 admits an (I, F-4)-partition and every planar graph with girth at least 8 admits an (I, F-2)-partition. The later result is best possible since neither girth condition nor the class of F-2 can be further improved. (C) 2018 Elsevier Inc. All rights reserved.