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General connectivity

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Graph Theory Singapore 1983

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1073))

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Abstract

Let X be a finite structure having property P, written X e P, and let Y be a nonempty subset of X. Then the general connectivity κ (X,Y : P) is the minimum cardinality of Z ⊂ Y such that X - Z ∉ P. Clearly graph connectivity (resp. line-connectivity) is the special case X=G=(V,E), Y=V(resp. E), and P=connected. Illustrations for groups, numbers, and graphs are given, including the following:

  1. (a)

    X=Y is a finite group and P means "generates X"; this case of general connectivity is easily characterized.

  2. (b)

    X=Y=Nn={1,2,...,n} and P means "contains a k-term arithmetic progression", as suggested by van der Waerden's Theorem.

  3. (c)

    X=Y=Nn again, but now P means that there exist three numbers x,y,z such that x+y=z. When x=y is permitted, this is reminiscent of Schur's existence theorem; otherwise Rado's theorem.

  4. (d)

    X=Kp, Y=E(Kp) and P means "contains Kn with n≦p". This is a reformulation of Turán's original problem in extremal graph theory.

  5. (e)

    X=G=(V,E), Y=E and P=hamiltonian. An extremal problem of this type was solved by Ore.

  6. (f)

    X=G is a connected graph, P=not graceful, and Y=V.

  7. (g)

    Again X=G and P=not graceful, but now Y=E. These graceful connectivities always exist, provided it is a true conjecture that all trees are graceful.

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Authors and Affiliations

  1. Department of Mathematics, The University of Michigan, 48109, Ann Arbor, MI, USA

    Frank Harary

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  1. Frank Harary
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Khee Meng Koh Hian Poh Yap

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© 1984 Springer-Verlag

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Harary, F. (1984). General connectivity. In: Koh, K.M., Yap, H.P. (eds) Graph Theory Singapore 1983. Lecture Notes in Mathematics, vol 1073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073107

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  • DOI: https://doi.org/10.1007/BFb0073107

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13368-1

  • Online ISBN: 978-3-540-38924-8

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