Degree-constrained 2-partitions of graphs

A $(\delta\geq k1,\delta\geq k2)$-partition of a graph $G$ is a vertex-partition $(V1,V2)$ of $G$ satisfying that $\delta(G[Vi])\geq ki$ for $i=1,2$. We determine, for all positive integers $k1,k2$, the complexity of deciding whether a given graph has a $(\delta\geq k1,\delta\geq k2)$-partition. We also address the problem of finding a function $g(k1,k2)$ such that the $(\delta\geq k1,\delta\geq k2)$-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 $k\ge 3$, that $g(2,2)=3$ and that $g(2,3)$, if it exists, has value 4 or 5.