An (F1, F4)-partition of graphs with low genus and girth at least 6

Let G = (V, E) be a graph. If the vertex set V (G) can be partitioned into two nonempty subsets V-1 and V-2 such that G[V-1] and G[V-2] are graphs with maximum degree at most d(1) and d(2), respectively, then we say that G has a (Delta(d1), Delta(d2))-partition. A similar definition can be given for the notation (F-d1, F-d2)-partition if G[V-i] is a forest with maximum degree at most d(i), where i is an element of {1, 2}. 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}. In this paper, we prove that every graph G with mad(G) <= 16/5 admits an (F-1, F-4)-partition. As a corollary, every graph with low genus and girth at least 6 admits an (F-1, F-4)-partition. This improves a result of Borodin and Kostochka saying that every graph G with mad(G) <= 16/5 admits a (Delta(1), Delta(4))-partition.