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A Better Bound on the Largest Induced Forests in Triangle-Free Planar Graph

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Abstract

It is well-known that there exists a triangle-free planar graph of n vertices that has the largest induced forest of order at most \(\frac{5n}{8}\). Salavatipour (Graphs Comb 22(1):113–126, 2006) proved that there is a forest of order at least \(\frac{5n}{9.41}\) in any triangle-free planar graph of n vertices. Dross et al. (Large induced forests in planar graphs with girth 4 or 5, arXiv:1409.1348, 2014) improved Salavatipour’s bound to \(\frac{5n}{9.17}\). In this work, we further improve the bound to \(\frac{5n}{9}\). Our technique is inspired by the recent ideas from Lukot’ka et al. (Electron J Comb 22(1):P1–P11, 2015).

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Notes

  1. We use lp_solve package. The full implementation can be found at the author’s homepage http://web.engr.oregonstate.edu/~lehu/res/lp_final.lp.

References

  1. Akiyama, J., Watanabe, M.: Maximum iduced forests of planar graphs. Graphs Comb. 1(3), 201–202 (1987)

    Article  Google Scholar 

  2. Albertson, M.O., Berman, D.M.: A conjecture on planar graphs. In: Bondy, J.A., Murty, U.S.R. (eds.) Graph Theory and Related Topics, p. 357. Academic Press (1979)

  3. Alon, N., Mubayi, D., Thomas, R.: Large induced forests in sparse graphs. J. Graph Theory 38(3), 113–123 (2001)

    Article  MathSciNet  Google Scholar 

  4. Borodin, O.V.: On acyclic colorings of planar graphs. Discrete Math. 25(3), 211–236 (1979)

    Article  MathSciNet  Google Scholar 

  5. Cranston, D.W., West, D.B.: A guide to the discharging method (2013). arXiv:1306.4434

  6. Dross, F., Montassier, M., Pinlou, A.: Large induced forests in planar graphs with girth 4 or 5 (2014). arXiv:1409.1348

  7. Kelly, T., Liu, C.: Size of the largest induced forest in subcubic graphs of girth at least four and five (2016). arXiv:1603.03855

  8. Kowalik, L., Lužar, B., Škrekovski, R.: An improved bound on the largest induced forests for triangle-free planar graphs. Discrete Math. Theor. Comput. Sci. 12(1), 87–100 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Lukot’ka, R., Mazák, J., Zhu, X.: Maximum 4-degenerate subgraph of a planar graph. Electron. J. Comb. 22(1), P1–P11 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Salavatipour, M.R.: Large induced forests in triangle-free planar graphs. Graphs Comb. 22(1), 113–126 (2006)

    Article  MathSciNet  Google Scholar 

  11. Wang, Y., Xie, Q., Yu, X.: Induced forests in bipartite planar graphs (2016). arXiv:1605.00047

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Acknowledgements

We thank Baigong Zheng for proofreading this paper. We thank conversations with Glencora Borradaile and Melissa Sherman-Bennett during the development of this work. We also would like to thank Bojan Mohar for pointing out mistakes in the statement of Theorem 1 in earlier versions of this paper. This material is based upon work supported by the National Science Foundation under Grant No. CCF-1252833.

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  1. School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, USA

    Hung Le

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  1. Hung Le
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Le, H. A Better Bound on the Largest Induced Forests in Triangle-Free Planar Graph. Graphs and Combinatorics 34, 1217–1246 (2018). https://doi.org/10.1007/s00373-018-1944-2

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