On connected degree sequences

TL;DR

This paper establishes precise conditions for when a sequence of non-negative integers can represent the degree sequence of a connected simple graph, providing both necessary and sufficient criteria and an alternative characterization.

Contribution

It offers a complete characterization of connected degree sequences, including a new alternative condition, clarifying a classical problem in graph theory.

Findings

Provides necessary and sufficient conditions for connected degree sequences.

Introduces a new alternative characterization of these conditions.

Clarifies a classical problem in graph theory.

Abstract

This note gives necessary and sufficient conditions for a sequence of non-negative integers to be the degree sequence of a connected simple graph. This result is implicit in a paper of Hakimi. A new alternative characterisation of these necessary and sufficient conditions is also given.