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The asymptotic connectivity of labelled coloured regular bipartite graphs

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1036)

Abstract

A labelled coloured bipartite graph, of LCBG, is a bipartite (simple) graph whose vertices have been 2-coloured and the vertices of each colour labelled independently. It is shown that for fixed r⩾3 the proportion of r-regular LCBGs on 2n vertices which are r-connected approaches 1 as n → ∞. Also, fix r⩾3 and q>0; let g=max(4,2{q/(2(r−2))}). Then the numbers of the following types of r-regular LCBGs with 2n vertices are asymptotically equal as n → ∞: those with girth at least g; those which are cyclically-q-edge-connected; and those which are cyclically-q-vertex-connected.

Keywords

  • Bipartite Graph
  • Vertex Degree
  • Degree Sequence
  • Label Graph
  • Black Vertex

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Edward A. Bender, The asymptotic number of non-negative integer matrices with given row and column sums, Discrete Math. 10, (1974), 217–223.

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  2. B.D. McKay and N.C. Wormald, Automorphisms of random graphs with specified degrees, preprint.

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  3. Nicholas C. Wormald, Some problems in the Enumeration of Labelled Graphs, Ph.D. Thesis, University of Newcastle, N.S.W. (1978).

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  4. Nicholas C. Wormald, The asymptotic connectivity of labelled regular graphs, J. Combin. Theory Ser. B, 31 (1981), 156–167.

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© 1983 Springer-Verlag

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Ellingham, M.N. (1983). The asymptotic connectivity of labelled coloured regular bipartite graphs. In: Casse, L.R.A. (eds) Combinatorial Mathematics X. Lecture Notes in Mathematics, vol 1036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071518

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  • DOI: https://doi.org/10.1007/BFb0071518

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12708-6

  • Online ISBN: 978-3-540-38694-0

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