Some remarks on the odd hadwiger’s conjecture

Abstract

We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic.

Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K k -minor is ck-colorable.

Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Böhme et al.

We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) − 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.

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

References

  1. [1]

    K. Appel and W. Haken: Every planar map is four colorable, Part I. Discharging; Illinois J. Math. 21 (1977), 429–490.

    MATH  MathSciNet  Google Scholar 

  2. [2]

    K. Appel, W. Haken and J. Koch: Every planar map is four colorable, Part II. Reducibility; Illinois J. Math. 21 (1977), 491–567.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    T. Böhme, K. Kawarabayashi, J. Maharry and B. Mohar: Linear connectivity forces large complete bipartite minors, to appear in J. Combin. Theory Ser. B.

  4. [4]

    P. A. Catlin: A bound on the chromatic number of a graph, Discrete Math 22 (1978), 81–83.

    Article  MathSciNet  Google Scholar 

  5. [5]

    R. Diestel: Graph Theory, Graduate texts in mathematics 173, Springer (1997).

  6. [6]

    G. A. Dirac: A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 85–92.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    P. Duchet and H. Meyniel: On Hadwiger’s number and the stability number, Annals of Discrete Math. 13 (1982), 71–74.

    MATH  MathSciNet  Google Scholar 

  8. [8]

    H. Hadwiger: Über eine Klassifikation der Streckenkomplexe, Vierteljahrsschr. Naturforsch, Ges. Zürich 88 (1943), 133–142.

    MathSciNet  Google Scholar 

  9. [9]

    J. Geelen, B. Gerards, L. Goddyn, B. Reed, P. Seymour and A. Vetta: The odd case of Hadwiger’s conjecture, submitted.

  10. [10]

    B. Guenin: Talk at Oberwolfach on Graph Theory, Jan. 2005.

  11. [11]

    T. R. Jensen and B. Toft: Graph Coloring Problems, Wiley-Interscience, 1995.

  12. [12]

    K. Kawarabayashi: Minors in 7-chromatic graphs, preprint.

  13. [13]

    K. Kawarabayashi and B. Toft: Any 7-chromatic graph has K 7 or K 4,4 as a minor, Combinatorica 25(3) (2005), 327–353.

    MATH  Article  MathSciNet  Google Scholar 

  14. [14]

    K. Kawarabayashi, M. Plummer and B. Toft: Improvements of the theorem of Duchet and Meyniel on Hadwiger’s Conjecture, J. Combin. Theory Ser. B 95 (2005), 152–167.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    K. Kawarabayashi and Z. Song: Independence number and clique minors, submitted.

  16. [16]

    A. Kostochka: The minimum Hadwiger number for graphs with a given mean degree of vertices (in Russian), Metody Diskret. Analiz. 38 (1982), 37–58.

    MATH  MathSciNet  Google Scholar 

  17. [17]

    A. Kostochka: Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica 4 (1984), 307–316.

    MATH  Article  MathSciNet  Google Scholar 

  18. [18]

    M. D. Plummer, M. Stiebitz and B. Toft: On a special case of Hadwiger’s Conjecture, Discuss. Math. Graph Theory 23 (2003), 333–363.

    MATH  MathSciNet  Google Scholar 

  19. [19]

    B. Reed and P. D. Seymour: Fractional colouring and Hadwiger’s conjecture, J. Combin. Theory Ser. B 74 (1998), 147–152.

    MATH  Article  MathSciNet  Google Scholar 

  20. [20]

    N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas: The four-color theorem, J. Combin. Theory Ser. B 70 (1997), 2–44.

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    N. Robertson, P. D. Seymour and R. Thomas: Hadwiger’s conjecture for K 6-free graphs, Combinatorica 13 (1993), 279–361.

    MATH  Article  MathSciNet  Google Scholar 

  22. [22]

    A. Thomason: An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 261–265.

    MATH  MathSciNet  Article  Google Scholar 

  23. [23]

    A. Thomason: The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001), 318–338.

    MATH  Article  MathSciNet  Google Scholar 

  24. [24]

    K. Wagner: Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), 570–590.

    MATH  Article  MathSciNet  Google Scholar 

  25. [25]

    D. R. Woodall: Subcontraction-equivalence and Hadwiger’s Conjecture, J. Graph Theory 11 (1987), 197–204.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ken-ichi Kawarabayashi.

Additional information

Research partly supported by the Japan Society for the Promotion of Science for Young Scientists, by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by Sumitomo Foundation and by Inoue Research Award for Young Scientists.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kawarabayashi, Ki., Song, ZX. Some remarks on the odd hadwiger’s conjecture. Combinatorica 27, 429 (2007). https://doi.org/10.1007/s00493-007-2213-9

Download citation

Mathematics Subject Classification (2000)

  • 05C15
  • 05C83