Characterization of potentially connected integer-pair sequences

Abstract

Given a pseudograph, multigraph or graph G, we can associate with it a sequence of unordered integer-pairs S_G=(c₁, c₂, …, c_q), where q=|E(G)|, constructed as follows: If the edges of G are labelled 1,2, …, q, then for the s^th edge (u,v) of G, c_s=(a_s, b_s) where a_s, b_s are the degrees of u and v. An integer-pair sequence S is said to be graphic if there exists a graph G for with S_G=S. In this paper we characterize potentially connected integer-pair sequences.