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Cycle Traversability for Claw-Free Graphs and Polyhedral Maps

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Abstract

Let G be a graph, and \(v \in V(G)\) and \(S \subseteq V(G)\setminus{v}\) of size at least k. An important result on graph connectivity due to Perfect states that, if v and S are k-linked, then a (k−1)-link between a vertex v and S can be extended to a k-link between v and S such that the endvertices of the (k−1)-link are also the endvertices of the k-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets S1 and S2 are k-linked, then a t-link (t<k) between two disjoint sets S1 and S2 can be extended to a k-link between S1 and S2 such that the endvertices of the t-link are preserved in the k-link.

Next, we are able to use these results to show that a 3-connected claw-free graph always has a cycle passing through any given five vertices, but avoiding any other one specified vertex. We also show that this result is sharp by exhibiting an infinite family of 3-connected claw-free graphs in which there is no cycle containing a certain set of six vertices but avoiding a seventh specified vertex. A direct corollary of our main result shows that a 3-connected claw-free graph has a topological wheel minor Wk with k ≤ 5 if and only if it has a vertex of degree at least k.

Finally, we also show that a graph polyhedrally embedded in a surface always has a cycle passing through any given three vertices, but avoiding any other specified vertex. The result is best possible in the sense that the polyhedral embedding assumption is necessary, and there are infinitely many graphs polyhedrally embedded in surfaces having no cycle containing a certain set of four vertices but avoiding a fifth specified vertex.

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References

  1. J. A. Bondy and L. Lovász: Cycles through specified vertices of a graph, Combinatorica1 (1981), 117–140.

    Article  MathSciNet  Google Scholar 

  2. Z. Chen: A twelve vertex theorem for 3-connected claw-free graphs, Graphs Combin.32 (2016), 553–558.

    Article  MathSciNet  Google Scholar 

  3. M. Chudnovsky and P. D. Seymour: The structure of claw-free graphs, in: Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005, 153–171.

    Google Scholar 

  4. V. Chvátal: New directions in Hamiltonian graph theory, (Proc. Third Ann Arbor Conf. Graph Theory, Univ. Michigan, Ann Arbor, Mich., 1971), Academic Press, New York, 1973, 65–95.

    Google Scholar 

  5. G. A. Dirac: In abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen, Math. Nachr.22 (1960), 61–85.

    Article  MathSciNet  Google Scholar 

  6. M. N. Ellingham, D. A. Holton and C. H.C. Little: Cycles through ten vertices in 3-connected cubic graphs, Combinatorica4 (1984), 265–273.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. E. Flandrin, E. Győri, H. Li and J. Shu: Cyclability in k-connected K1,4-free graphs, Discrete Math.310 (2010), 2735–2741.

    Article  MathSciNet  Google Scholar 

  9. R. Gould: A look at cycles containing specified elements of a graph, Discrete Math.309 (2009), 6299–6311.

    Article  MathSciNet  Google Scholar 

  10. E. Győri and M. D. Plummer: A nine vertex theorem for 3-connected claw-free graphs, Stud. Sci. Math. Hungar.38 (2001), 233–244.

    MathSciNet  MATH  Google Scholar 

  11. R. Häggkvist and W. Mader: Circuits through prescribed vertices in k-connected k-regular graphs, J. Graph Theory39 (2002), 145–163.

    Article  MathSciNet  Google Scholar 

  12. R. Häggkvist and C. Thomassen: Circuits through specified edges, Discrete Math.41 (1982), 29–34.

    Article  MathSciNet  Google Scholar 

  13. R. Halin: Zur Theorie der n-fach zusammenhängenden Graphen, Abh. Math. Sem Hamburg33 (1969), 133–164.

    Article  Google Scholar 

  14. D. A. Holton, B. D. McKay, M. D. Plummer and C. Thomassen: A nine point theorem for 3-connected graphs, Combinatorica2 (1982), 53–62.

    Article  MathSciNet  Google Scholar 

  15. D. A. Holton and M. D. Plummer: Cycles through prescribed and forbidden point sets, (Workshop on Combinatorial Optimization, Bonn, 1980), Ann. Discrete Math.16, North-Holland, 1982, 129–147.

    MATH  Google Scholar 

  16. D. A. Holton and C. Thomassen: Research problem 81, Discrete Math.62 (1986), 111–112.

    Article  Google Scholar 

  17. K. Kawarabayashi: One or two disjoint circuits cover independent edges: Lovász—Woodall Conjecture, J. Combin. Theory Ser. B84 (2002), 1–44.

    Article  MathSciNet  Google Scholar 

  18. K. Kawarabayashi: Cycles through a prescribed vertex set in N-connected graphs, J. Combin. Theory Ser. B90 (2004), 315–323.

    Article  MathSciNet  Google Scholar 

  19. A. K. Kelmans and M. V. Lomonosov: When m vertices in a k-connected graph cannot be walked round along a simple cycle, Discrete Math.38 (1982), 317–322.

    Article  MathSciNet  Google Scholar 

  20. L. Lovász, Problem 5, Per. Math. Hungar.4 (1974), 82.

    Google Scholar 

  21. M. M. Matthews and D. P. Sumner: Hamiltonian results in K1;3-free graphs, J. Graph Theory8 (1984), 139–146.

    Article  MathSciNet  Google Scholar 

  22. D. M. Mesner and M. E. Watkins: Some theorems about n-vertex connected graphs, J. Math. Mech.16 (1966), 321–326.

    MathSciNet  MATH  Google Scholar 

  23. B. Mohar and C. Thomassen: Graphs on Surfaces, Hopkins Univ. Press, Baltimore, 2001.

    MATH  Google Scholar 

  24. D. A. Nelson: Hamiltonian Graphs, M.A. Thesis, Vanderbilt University, 1973.

    Google Scholar 

  25. H. Perfect: Applications of Menger's Graph Theorem, J. Math. Anal. Appl.22 (1968), 96–111.

    Article  MathSciNet  Google Scholar 

  26. M. D. Plummer and E. Wilson: On cycles and connectivity in planar graphs, Canad. Math. Bull.16 (1973), 283–288.

    Article  MathSciNet  Google Scholar 

  27. G. T. Sallee: Circuits and paths through specified nodes, J. Combin. Theory Ser. B15 (1973), 32–39.

    Article  MathSciNet  Google Scholar 

  28. R. Thomas and X. Yu: 4-connected projective-planar graphs are Hamiltonian, J. Combin. Theory Ser. B62 (1994), 114–132.

    Article  MathSciNet  Google Scholar 

  29. W. T. Tutte: A theorem on planar graphs, Trans. Amer. Math. Soc.82 (1956), 99–116.

    Article  MathSciNet  Google Scholar 

  30. M. E. Watkins and D. M. Mesner: Cycles and connectivity in graphs, Canad. J. Math.19 (1967), 1319–1328.

    Article  MathSciNet  Google Scholar 

  31. D. R. Woodall: Circuits containing specified edges, J. Combin. Theory Ser. B22 (1977), 274–278.

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank the referees for their valuable comments.

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Correspondence to Dong Ye.

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Partially supported by the NKFIH Grant K116769.

Partially supported by a grant from the Simons Foundation (No. 359516).

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Győri, E., Plummer, M.D., Ye, D. et al. Cycle Traversability for Claw-Free Graphs and Polyhedral Maps. Combinatorica 40, 405–433 (2020). https://doi.org/10.1007/s00493-019-4042-z

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  • DOI: https://doi.org/10.1007/s00493-019-4042-z

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