Solution to a Forcible Version of a Graphic Sequence Problem

Abstract

Let $A_n=(a_1,a_2,\ldots,a_n)$ and $B_n=(b_1,b_2,\ldots,b_n)$ be nonnegative integer sequences with $A_n\le B_n$. The purpose of this note is to give a good characterization such that every integer sequence $\pi=(d_1,d_2,\ldots d_n)$ with even sum and $A_n\le \pi\le B_n$ is graphic. This solves a forcible version of problem posed by Niessen and generalizes the Erd\H{o}s--Gallai theorem.