Unfriendly partitions for graphs not containing a subdivision of an infinite clique

Abstract

We prove that in any graph containing no subdivision of an infinite clique there exists a partition of the vertices into two parts, satisfying the condition that every vertex has at least as many neighbors in the part not containing it as it has in its own part.

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

References

  1. [1]

    R. Aharoni, E. C. Milner and K. Prikry: Unfriendly partitions of a graph, Jour. Combin. Theo. Sir. B. 50 (1990), 1–10.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    H. Bruhn, R. Diestel, A. Georgakopoulos and P. Sprüssel: Every rayless graph has an unfriendly partition, Combinatorica 30 (2010), 521–532.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    R. Cowan and W. Emerson: Proportional colorings of graphs, unpublished.

  4. [4]

    F. Galvin: Indeterminacy of point-open games, Bull. Acad. Polon. Sci., Ser. Math. 26 (1978), 445–449.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    E. C. Milner and S. Shelah: Graphs with no unfriendly partitions, in: A tribute to Paul Erdös, 373–384, Cambridge Univ. Press, Cambridge, 1990.

    Google Scholar 

  6. [6]

    N. Robertson, P. D. Seymour and R. Thomas: Excluding subdivisions of infinite cliques, Trans. Amer. Math. Soc. 332 (1992), 211–223.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Eli Berger.

Additional information

The research was supported by BSF grant no. 2006099 and by an ISF grant.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Berger, E. Unfriendly partitions for graphs not containing a subdivision of an infinite clique. Combinatorica 37, 157–166 (2017). https://doi.org/10.1007/s00493-015-3261-1

Download citation

Mathematics Subject Classification (2000)

  • 05C63