Stability Results on the Circumference of a Graph

Abstract

In this paper, we extend and refine previous Turán-type results on graphs with a given circumference. Let Wn,k,c be the graph obtained from a clique Kck+1 by adding n − (ck + 1) isolated vertices each joined to the same k vertices of the clique, and let f(n,k,c) = e(Wn,k,c). Improving a celebrated theorem of Erdős and Gallai [8], Kopylov [18] proved that for c<n, any 2-connected graph G on n vertices with circumference c has at most \(\max \left\{f(n, 2, c), f\left(n,\left\lfloor\frac{c}{2}\right\rfloor, c\right)\right\}\) edges, with equality if and only if G is isomorphic to Wn,2,c or \({W_{n,2,c}}\). Recently, Füredi et al. [15,14] proved a stability version of Kopylov’s theorem. Their main result states that if G is a 2-connected graph on n vertices with circumference c such that 10 ≤ c<n and \(W_{n,\left\lfloor\frac{c}{2}\right\rfloor, c}\), then either G is a subgraph of Wn,2,c or \(e\left( G \right) > \max \left\{ {f(n,3,c),f\left( {n,\left\lfloor {{c \over 2},} \right\rfloor - 1,c} \right)} \right\}\), or c is odd and G is a subgraph of a member of two well-characterized families which we define as \(e(G)>\max \left\{f(n, k+1, c), f\left(n,\left\lfloor\frac{c}{2}\right\rfloor-1, c\right)\right\}\) and \(W_{n,\left\lfloor\frac{c}{2}\right\rfloor, c}\).

We prove that if G is a 2-connected graph on n vertices with minimum degree at least k and circumference c such that 10 ≤ c<n and \(\mathcal{X}_{n, c} \cup \mathcal{Y}_{n, c}\), then one of the following holds:

  1. (i)

    G is a subgraph of Wn,k,c or \({W_{n,2,c}}\)

  2. (ii)

    k = 2, c is odd, and G is a subgraph of a member of \(W_{n,\left\lfloor\frac{c}{2}\right\rfloor, c}\), or

  3. (iii)

    k ≤ 3 and G is a subgraph of the union of a clique Kck+1 and some cliques Kk+1’s, where any two cliques share the same two vertices.

This provides a unified generalization of the above result of Füredi et al. [15,14] as well as a recent result of Li et al. [20] and independently, of Füredi et al. [12] on non-Hamiltonian graphs. graphs. A refinement and some variants of this result are also obtained. Moreover, we prove a stability result on a classical theorem of Bondy [2] on the circumference. We use a novel approach, which combines several proof ideas including a closure operation and an edge-switching technique. We will also discuss some potential applications of this approach for future research.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Bollobás: Extremal Graph Theory, Academic Press, New York (1978).

    Google Scholar 

  2. [2]

    J. A. Bondy: Large cycles in graphs, Discrete Math.1 1971/1972, no. 2, 121–132.

    MathSciNet  Article  Google Scholar 

  3. [3]

    J. A. Bondy and V. Chvátal: A method in graph theory, Discrete Math.15 (1976), 111–135.

    MathSciNet  Article  Google Scholar 

  4. [4]

    J. A. Bondy and U. S. R. Murty: Graph Theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008. xii+651 pp. ISBN: 978-1-84628-969-9.

    Google Scholar 

  5. [5]

    V. Chvátal: On Hamilton’s ideals, J. Combin. Theory Ser. B12 (1972), 163–168.

    MathSciNet  Article  Google Scholar 

  6. [6]

    G. A. Dirac: Some theorems on abstract graphs, Proc. London Math. Soc.(3-2) (1952), 69–81.

    MathSciNet  Article  Google Scholar 

  7. [7]

    P. Erdős: Remarks on a paper of Pósa, Magyar Tud. Akad. Mat. Kutató Int. Közl.7 (1962), 227–229.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    P. Erdős and T. Gallai: On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar.10 (1959), 337–356.

    MathSciNet  Article  Google Scholar 

  9. [9]

    G. Fan: Long cycles and the codiameter of a graph I, J. Combin. Theory Ser. B49 (1990), 151–180.

    MathSciNet  Article  Google Scholar 

  10. [10]

    G. Fan, X. Lv and P. Wang: Cycles in 2-connected graphs, J. Combin. Theory Ser. B92 (2004), 379–394.

    MathSciNet  Article  Google Scholar 

  11. [11]

    R. J. Faudree and R. H. Schel: Path Ramsey numbers in multiclorings, J. Combin. Theory Ser. B19 (1975), 150–160.

    Article  Google Scholar 

  12. [12]

    Z. Füredi, A. Kostochka and R. Luo: A stability version for a theorem of Erdős on nonhamiltonian graphs, Discrete Math.340 (2017), 2688–2690.

    MathSciNet  Article  Google Scholar 

  13. [13]

    Z. Füredi, A. Kostochka and R. Luo: Extensions of a theorem of Erdős on non-hamiltonian graphs, J. Graph Theory89 (2018), 176–193.

    MathSciNet  Article  Google Scholar 

  14. [14]

    Z. Füredi, A. Kostochka, R. Luo and J. Verstraëte: Stability in the Erdős—Gallai Theorem on cycles and paths, II, Discrete Math.341 (2018), 1253–1263.

    MathSciNet  Article  Google Scholar 

  15. [15]

    Z. Füredi, A. Kostochka and J. Verstraëte: Stability in the Erdős—Gallai theorems on cycles and paths, J. Combin. Theory Ser. B121 (2016), 197–228.

    MathSciNet  Article  Google Scholar 

  16. [16]

    Z. Füredi and M. Simonovits: The history of degenerate (bipartite) extremal graph problems, Bolyai Math. Studies25, 169–264, in: Erdős Centennial (L. Lovász, I. Ruzsa and V. T. Sós, Eds.) Springer, 2013. Also see arXiv:1306.5167.

  17. [17]

    D. König: Graphs and matrices, Mat. Fiz. Lapok38 (1931), 116–119 (in Hugarian).

    MATH  Google Scholar 

  18. [18]

    G. N. Kopylov: Maximal paths and cycles in a graph, Dokl. Akad. Nauk SSSR234 (1977), 19–21.

    MathSciNet  Google Scholar 

  19. [18a]

    G. N. Kopylov: (English translation: Soviet Math. Dokl. 18 (1977), 593–596.)

    Google Scholar 

  20. [19]

    M. Lewin: On maximal circuits in directed graphs, J. Combin. Theory Ser. B18 (1975), 175–179.

    MathSciNet  Article  Google Scholar 

  21. [20]

    B. Li and B. Ning: Spectral analogues of Erdős’ and Moon-Moser’s theorems on Hamilton cycles, Linear Multilinear Algebra64 (2016), 2252–2269.

    MathSciNet  Article  Google Scholar 

  22. [21]

    O. Ore: Arc coverings of graphs, Ann. Mat. Pura Appl.55 (4) (1961) 315–321.

    MathSciNet  Article  Google Scholar 

  23. [22]

    Z. Ryjáček: On a closure concept in claw-free graphs, J. Combin. Theory Ser. B70 (1997), 217–224.

    MathSciNet  Article  Google Scholar 

  24. [23]

    D. R. Woodall: Maximal circuits of graphs I, Acta Math. Acad. Sci. Hungar.28 (1976), 77–80.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgement

The first author would like to thank Alexandr V. Kostochka for helpful discussions.

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Jie Ma or Bo Ning.

Additional information

Research supported in part by National Natural Science Foundation of China grants 11501539 and 11622110 and Anhui Initiative in Quantum Information Technologies grant AHY150200.

Research supported in part by National Natural Science Foundation of China grant 11601379.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ma, J., Ning, B. Stability Results on the Circumference of a Graph. Combinatorica 40, 105–147 (2020). https://doi.org/10.1007/s00493-019-3843-4

Download citation

Mathematics Subject Classification (2010)

  • 05C35
  • 05C38
  • 05D99