Some criteria for integer sequences pair being realizable by a graph

TL;DR

This paper investigates criteria for when pairs of integer sequences can be realized as degree sequences of a simple graph, extending classical characterizations by Berge and Ryser.

Contribution

It generalizes six classical characterizations of degree sequence pairs to broader conditions and provides new related results.

Findings

Established criteria for realizability of degree sequence pairs.

Extended classical theorems to more general sequence conditions.

Provided new characterizations related to graph degree sequences.

Abstract

Let $A=(a_1,\ldots,a_n)$ and $B=(b_1,\ldots,b_n)$ be two sequences of nonnegative integers with $a_i \le b_i$ for $1\le i\le n$. The pair $(A;B)$ is said to be realizable by a graph if there exists a simple graph $G$ with vertices $v_1,\ldots, v_n$ such that $a_i\le d_G(v_i)\le b_i$ for $1\le i\le n$. Let $\preceq$ denote the lexicographic ordering on $Z\times Z:$ $(a_{i+1},b_{i+1})\preceq (a_i,b_i)\Longleftrightarrow [(a_{i+1}<a_i)\vee ((a_{i+1}=a_i)\&(b_{i+1}\le b_i))]$. We say that the sequences $A$ and $B$ are in good order if $(a_{i+1},b_{i+1})\preceq (a_i,b_i)$. In this paper, we consider the generalizations of six classical characterizations on sequences pair due to Berge, Ryser et al. and present related results.