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An extremal problem for cycles in hamiltonian graphs

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Abstract

For integersp andr with 3 ≤r ≤ p − 1, letf(p, r) denote the maximum number of edges in a hamiltonian graph of orderp which does not contain a cycle of lengthr. Results from literature on the determination off(p, r) are collected and a number of new lower bounds, many of which are conjectured to be best possible, are given. The main result presented is the proof thatf(p, 5) = (p − 3)2/4 + 5 for oddp ≥ 11.

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References

  1. Bondy, J.A.: Large cycles in graphs. Discrete Math.1, 121–132 (1971)

    Google Scholar 

  2. Bondy, J.A.: Pancyclic graphs. J. Comb. Theory Ser.B 11, 80–84 (1971)

    Google Scholar 

  3. Bondy, J.A., Simonovits, M.: Cycles of even lengths in graphs, J. Comb. Theory Ser.B 16, 97–105 (1974)

    Google Scholar 

  4. Brandt, S., Faudree, R.J., Goddard, W.: Weakly pancyclic graphs, in preparation

  5. Clapham, C.R.J., Flockhart, A., Sheehan, J.: Graphs without four-cycles. J. Graph Theory13, 29–47 (1989)

    Google Scholar 

  6. Füredi, Z.: Graphs without quadrilaterals. J. Comb. Theory Ser.B 34, 187–190 (1983)

    Google Scholar 

  7. Häggkvist, R.: Odd cycles of specified length in non-bipartite graphs. Ann. Discr. Math.13, 89–99 (1982)

    Google Scholar 

  8. Häggkvist, R., Faudree, R.J., Schelp, R.H.: Pancyclic graphs — connected Ramsey number. Ars Comb.11, 37–49 (1981)

    Google Scholar 

  9. Lesniak-Foster, L.: On critically hamiltonian graphs. Acta Math. Acad. Sci. Hungar.29, 255–258 (1977)

    Google Scholar 

  10. Woodall, D.R.: Sufficient conditions for circuits in graphs, Proc. London Math. Soc. (3)24, 739–755 (1972)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Science, The Edward Wright Building, University of Aberdeen, Dunbar Street, AB9 2TY, Aberdeen, Scotland

    George R. T. Hendry

  2. Graduiertenkolleg “Algorithmishe Diskrete Mathematik”, FB Mathematik, Freie Universität Berlin, Arnimallee 2-6, 14195, Berlin, Germany

    Stephan Brandt

Authors
  1. George R. T. Hendry
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  2. Stephan Brandt
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Additional information

George Hendry died during the publication process.

Supported by Deutsche Forschungsgemeinschaft (DFG), Grant We 1265.

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Hendry, G.R.T., Brandt, S. An extremal problem for cycles in hamiltonian graphs. Graphs and Combinatorics 11, 255–262 (1995). https://doi.org/10.1007/BF01793012

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  • DOI: https://doi.org/10.1007/BF01793012

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