A Characterization of Box-bounded Degree Sequences of Graphs

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 $A_n\le B_n$ and $b_i\ge b_{i+1},a_i+b_i\ge a_{i+1}+b_{i+1}, i=1,2\ldots , n-1$. The purpose of this note is to give a good characterization for $A_n$ and $B_n$ 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 improves related results of Guo and Yin and generalizes the Erdős–Gallai theorem.