Degree-constrained 2-partitions of graphs

A (δ≥ k1,δ≥ k2)-partition of a graph G is a vertex-partition (V1,V2) of G satisfying that δ(G[Vi])≥ ki for i=1,2. We determine, for all positive integers k1,k2, the complexity of deciding whether a given graph has a (δ≥ k1,δ≥ k2)-partition. We also address the problem of finding a function g(k1,k2) such that the (δ≥ k1,δ≥ k2)-partition problem is cal NP-complete for the class of graphs of minimum degree less than g(k1,k2) and polynomial for all graphs with minimum degree at least g(k1,k2). We prove that g(1,k)=k for k≥ 3, that g(2,2)=3 and that g(2,3), if it exists, has value 4 or 5.