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Hamiltonian paths in infinite graphs

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Abstract

A tight connection is exhibited between infinite paths in recursive trees and Hamiltonian paths in recursive graphs. A corollary is that determining Hamiltonicity in recursive graphs is highly undecidable, viz, Σ 11 -complete. This is shown to hold even for highly recursive graphs with degree bounded by 3. Hamiltonicity is thus an example of an interesting graph problem that is outside the arithmetic hierarchy in the infinite case.

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References

  1. R. Aharoni, M. Magidor and R. A. Shore,On the strength of König’s duality theorem for infinite bipartite graphs, manuscript, 1990.

  2. D. R. Bean,Effective coloration, J. Sym. Logic41 (1976), 469–480.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. R. Bean,Recursive Euler and Hamiltonian paths, Proc. Am. Math. Soc.55 (1976), 385–394.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Beigel and W. I. Gasarch, unpublished results, 1986–1990.

  5. R. Beigel and W. I. Gasarch,On the complexity of finding the chromatic number of a recursive graph, Parts I & II, Ann. Pure and Appl. Logic45 (1989), 1–38, 227–247.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. A. Burr,Some undecidable problems involving the edge-coloring and vertex coloring of graphs, Disc. Math.50 (1984), 171–177.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. I. Gasarch and M. Lockwood,The existence of matchings for recursive and highly recursive bipartite graphs, Technical Report 2029, Univ. of Maryland, May 1988.

  8. W. I. Gasarch, Personal communication, 1991.

  9. D. Harel,Effective transformations on infinite trees, with applications to high undecidability, dominoes and fairness, J. Assoc. Comput. Mach.33 (1986), 224–248.

    MathSciNet  Google Scholar 

  10. A. Manaster and J. Rosenstein,Effective matchmaking (recursion theoretic aspects of a theorem of Philip Hall), Proc. London Math. Soc.3 (1972), 615–654.

    Article  MathSciNet  Google Scholar 

  11. A. Nerode and J. Remmel,A survey of lattices of R. E. substructures, in Recursion Theory, Proc. Symp. in Pure Math., Vol. 42 (A. Nerode and R. A. Shore, eds.), Am. Math. Soc., Providence, R. I., 1985, pp. 323–375.

    Google Scholar 

  12. H. Rogers,Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot, Israel

    David Harel

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  1. David Harel
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Correspondence to David Harel.

Additional information

Parts of this research were carried out during a visit to IBM T.J. Watson Research Center, Hawthorne, NY, in the Summer of 1990. The author holds the William Sussman Professorial Chair in Mathematics.

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Harel, D. Hamiltonian paths in infinite graphs. Israel J. Math. 76, 317–336 (1991). https://doi.org/10.1007/BF02773868

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  • DOI: https://doi.org/10.1007/BF02773868

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