Parallel enumeration of degree sequences of simple graphs. II.

Abstract In the paper we report on the parallel enumeration of the degree sequences (their number is denoted by G(n)) and zerofree degree sequences (their number is denoted by (Gz(n)) of simple graphs on n = 30 and n = 31 vertices. Among others we obtained that the number of zerofree degree sequences of graphs on n = 30 vertices is Gz(30) = 5 876 236 938 019 300 and on n = 31 vertices is Gz(31) = 22 974 847 474 172 374. Due to Corollary 21 in [52] these results give the number of degree sequences of simple graphs on 30 and 31 vertices.</jats:p

Characterization of unigraphic and unidigraphic integer-pair sequences

AbstractGiven a graph (digraph) G with edge (arc) set E(G) = {(u1}, υ1), (u2, υ2),⋯,(uq, υq, where q = |E(G)|, we can associate with it an integer-pair sequence SG = ((a1, b1), (a2, b2),⋯, (aq, bq)) where ai, bi are the degrees (indegrees) of ui, υi respectively. An integer- pair sequence S is said to be graphic (digraphic) if there exists a graph (digraph) G such that SG = S. In this paper we characterize unigraphic and unidigraphic integer-pair sequences