Skip to main content

Connected subgraphs of the graph of multigraphic realisations of a degree sequence

Contributed Papers

Part of the Lecture Notes in Mathematics book series (LNM,volume 884)

Abstract

An m-graph is a graph, without loops, but with multiple edges of any multiplicity less than or equal to m. An exact m-graph is an m-graph with at least one edge of multiplicity m. A new proof is given that the graph \(R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))\), of all m-graphic realisations of a degree sequence, \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d}\), is connected. This is done by taking any two vertices of \(R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))\), say G and H, and finding a path between them which preserves any previously chosen edge of multiplicity m that occurs in both G and H. The construction of this path also establishes best possible upper and lower bounds on the length of the shortest path between any two vertices of \(R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))\).

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V. Chungphaisan, Conditions for sequences to be r-graphic, Discrete Math. 7 (1974), 31–39.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. D.R. Fulkerson, A.J. Hoffman, and M.H. McAndrew, Some properties of graphs with multiple edges, Canad. J. Math. 17 (1965), 166–177.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1981 Springer-Verlag

About this paper

Cite this paper

Billington, D. (1981). Connected subgraphs of the graph of multigraphic realisations of a degree sequence. In: McAvaney, K.L. (eds) Combinatorial Mathematics VIII. Lecture Notes in Mathematics, vol 884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091814

Download citation

  • DOI: https://doi.org/10.1007/BFb0091814

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10883-2

  • Online ISBN: 978-3-540-38792-3

  • eBook Packages: Springer Book Archive