Degree-constrained 2-partitions of graphs

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A (δ≥k1,δ≥k2)-partition of a graph G is a vertex-partition (V1,V2) of G into two non-empty sets 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 NP-complete for the class of graphs of minimum degree less than g(k1,k2) and polynomial time solvable for all graphs with minimum degree at least g(k1,k2). We prove that g(1,k) exists and has value k for all k≥3, that g(2,2) also exists and has value 3 and that g(2,3), if it exists, has value 4 or 5.