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.