Partitions of multigraphs under minimum degree constraints

In 1996, Michael Stiebitz proved that if G is a simple graph with delta(G) >= s+t+1 and s, t is an element of Z(0), then V(G) can be partitioned into two sets A and B such that delta(G[A]) >= s and delta(G[B]) >= t. In 2016, Amir Ban proved a similar result for weighted graphs. Let G be a simple graph with at least two vertices, let w : E(G) -> R->0 be a weight function, let s, t is an element of R->= 0, and let W = max(e is an element of E)(G) w(e). If delta(G) >= s + t + 2W, then V(G) can be partitioned into two sets A and B such that delta(G[A]) >= s and delta(G[B]) >= t. This motivated us to consider this partition problem for multigraphs, or equivalently for weighted graphs (G, w) with w : E(G) -> Z(>= 1). We prove that ifs, t is an element of Z(>= 0) and delta(G) >= s+t+2W-1 >= 1, then V(G) can be partitioned into two sets A and B such that delta(G[A]) >= s and delta(G[B]) >= t. We also prove a variable version of this result and show that for K-4(-)-free graphs, the bound on the minimum degree can be decreased. (C) 2018 Elsevier B.V. All rights reserved.