Solution to a problem on degree sequences of graphs

Let a1,a2,…,an and b1,b2,…,bn be integers with 0⩽ai⩽bi for i=1,2,…,n. The purpose of this note is to give a good characterization for the existence of a simple graph G with vertices v1,v2,…,vn such that ai⩽dG(vi)⩽bi for i=1,2,…,n. This solves a research problem posed by Niessen and generalizes an Erdős–Gallai theorem.