Enumeration of the degree sequences of non-separable graphs and connected graphs

Øystein J. Rødseth, James Allen Sellers, Helge Tverberg

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d1, d2, ..., dn to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions dn s (2 m) and dc (2 m), the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: dn s (2 m) = p (2 m) - p (2 m - 1) - underover(∑, j = 0, m - 2) p (j) and dc (2 m) = p (2 m) - p (m - 1) - 2 underover(∑, j = 0, m - 2) p (j) where p (j) is the number of unrestricted integer partitions of j.

Original languageEnglish (US)
Pages (from-to)1309-1317
Number of pages9
JournalEuropean Journal of Combinatorics
Volume30
Issue number5
DOIs
StatePublished - Jul 1 2009

Fingerprint

Degree Sequence
Nonseparable
Enumeration
Connected graph
Integer Partitions
Degree Sum
Bijective
Graph in graph theory
Generating Function
Necessary Conditions
Closed
Integer
Sufficient Conditions

All Science Journal Classification (ASJC) codes

  • Computational Theory and Mathematics
  • Geometry and Topology
  • Theoretical Computer Science

Cite this

Rødseth, Øystein J. ; Sellers, James Allen ; Tverberg, Helge. / Enumeration of the degree sequences of non-separable graphs and connected graphs. In: European Journal of Combinatorics. 2009 ; Vol. 30, No. 5. pp. 1309-1317.
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Enumeration of the degree sequences of non-separable graphs and connected graphs. / Rødseth, Øystein J.; Sellers, James Allen; Tverberg, Helge.

In: European Journal of Combinatorics, Vol. 30, No. 5, 01.07.2009, p. 1309-1317.

Research output: Contribution to journalArticle

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