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Theory and Applications of Graphs pp 286–303Cite as

The cartesian product of two graphs is stable

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 642))

Abstract

If G and H are two connected non-trivial graphs, not necessarily of finite order, then we show that the cartesian product G × H, is stable in the sense of Sheehan [4]. Moreover except when G=P2 and H is a certain restricted class of prime graphs, any edge of G × H may be removed to give the stability, i.e. G × H is completely stable.

Consideration is given to the case where G × H is not connected. In the case of finite graphs G × H is stable unless one of G or H is totally disconnected and the other is not stable. For non-finite graphs the situation is not as clear. We give examples of cartesian products of non-finite graphs which are not stable.

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References

  1. M. Behzad and G. Chartrand, Introduction To The Theory of Graphs, Allyn and Bacon, Boston, (1971).

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  2. W. Dörfler, Some results on the reconstruction of graphs, Coll. Math. Soc. János Bolyai, 10, North Holland, Amsterdam, (1975), 361–383.

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  3. G. Sabidussi, Graph Multiplication, Math. Zeit., 72, (1960), 446–457.

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  4. J. Sheehan, Fixing subgraphs, J. Comb. Th. 12(B), (1972), 226–244.

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  5. J. Sims and D. A. Holton, Stability of cartesian products, to appear.

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  6. J. Sims. Stability of the cartesian product of graphs, M.Sc. Thesis, University of Melbourne, 1976.

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Author information

Authors and Affiliations

  1. University of Melbourne, Parkville, Victoria, 3052, Australia

    D. A. Holton

  2. Linacre College, Oxford, England, OX1 1SY

    J. Sims

Authors
  1. D. A. Holton
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  2. J. Sims
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Yousef Alavi  & Don R. Lick  & 

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© 1978 Springer-Verlag Berlin Heidelberg

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Holton, D.A., Sims, J. (1978). The cartesian product of two graphs is stable. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070385

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  • DOI: https://doi.org/10.1007/BFb0070385

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08666-6

  • Online ISBN: 978-3-540-35912-8

  • eBook Packages: Springer Book Archive

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