Graph Decompositions Satisfying Extremal Degree Constraints

We show that the vertex set of any graph G with p ⩾2 vertices can be partitioned into non‐empty sets V 1 , V 2 , such that the maximum degree of the induced subgraph 〈 V i 〉 does not exceed\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{{p_i - 1}}{{p - 1}}\Delta (G), $$\end{document}where p i = | V i |, for i =1, 2. Furthermore, the structure of the induced subgraphs is investigated in the extreme case.