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Induced Cycles in Graphs

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Abstract

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by \(c_\mathrm{ind}(G)\). We prove that if G is an r-regular graph of order n, then \(c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}\) and we prove that if G is a cubic, claw-free graph on order n, then \(c_\mathrm{ind}(G) > \frac{13}{20}n\) and this bound is asymptotically best possible.

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References

  1. Cardoso, D.M., Kamiński, M., Lozin, V.: Maximum \(k\)-regular induced subgraphs. J. Comb. Optim. 14, 455–463 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Erdős, P.: On some of my favourite problems in various branches of combinatorics. Proceedings of the 4th Czechoslovakian symposium on combinatorics, graphs and complexity, practice. Ann. Discrete Math. 51(1992), 69–79 (1990)

  3. Flandrin, E., Faudree, R., Ryjáček, Z.: Claw-free graphs—a survey. Discrete Math. 164, 87–147 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Henning, M.A., Löwenstein, C.: Locating-total domination in claw-free cubic graphs. Discrete Math. 312, 3107–3116 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lovász, L., Plummer, M.D.: Matching Theory. In: Annals of Discrete Mathematics, vol. 29. North-Holland, Amsterdam (1986)

  7. Lozin, V., Mosca, R., Purcell, C.: Sparse regular induced subgraphs in \(2P_3\)-free graphs. Discrete Optim. 10, 304–309 (2013)

    Article  MathSciNet  Google Scholar 

  8. Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. J. Discrete Algorithms 7, 181–190 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Plummer, M.: Factors and factorization. In: Gross, J.L., Yellen, J. (eds.) Handbook of Graph Theory, pp. 403–430. CRC Press, New York (2003). ISBN: 1-58488-092-2

    Google Scholar 

  10. Pulleyblank, W.R.: Matchings and extension. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, pp. 179–232. Elsevier Science B.V, Amsterdam (1995). (ISBN 0-444-82346-8)

    Google Scholar 

  11. Tutte, W.T.: A short proof of the factor theorem for finite graphs. Can. J. Math. 6, 347–352 (1954)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Research of Michael A. Henning supported in part by the South African National Research Foundation and the University of Johannesburg.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa

    Michael A. Henning

  2. Institute of Optimization and Operations Research, Ulm University, 89081, Ulm, Germany

    Felix Joos, Christian Löwenstein & Thomas Sasse

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  1. Michael A. Henning
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  2. Felix Joos
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  3. Christian Löwenstein
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  4. Thomas Sasse
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Correspondence to Christian Löwenstein.

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Henning, M.A., Joos, F., Löwenstein, C. et al. Induced Cycles in Graphs. Graphs and Combinatorics 32, 2425–2441 (2016). https://doi.org/10.1007/s00373-016-1713-z

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  • DOI: https://doi.org/10.1007/s00373-016-1713-z

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