Skip to main content
SpringerLink
Log in

The Chromatic Symmetric Functions of Trivially Perfect Graphs and Cographs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distinguish rooted trees. In this paper, using a similar method, we prove that the chromatic symmetric functions distinguish trivially perfect graphs. Moreover, we also prove that claw-free cographs, that is, \( (K_{1,3},P_{4}) \)-free graphs belong to a known class of e-positive graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Chvátal, V., Hammer, P.L.: Aggregation of Inequalities in Integer Programming. Ann. Discret. Math. 1, 145–162 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Corneil, D.G., Lerchs, H., Burlingham, L.S.: Complement reducible graphs. Discrete Appl. Math. 3(3), 163–174 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cho, S., van Willigenburg, S.: Chromatic bases for symmetric functions. Electron. J. Combin. 23(1), P1.15 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Ésik, Z.: Free De Morgan bisemigroups and bisemilattices. Algebra Colloquium 10(1), 23–32 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gasharov, V.: Incomparability graphs of (3 + 1)-free posets are s-positive. Discrete Math. 157(1), 193–197 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gessel, I.M.: Multipartite P-partitions and inner products of skew Schur functions. Contemp. Math 34(289–301), 101 (1984)

    MathSciNet  MATH  Google Scholar 

  7. Golumbic, M.C.: Trivially perfect graphs. Discrete Math. 24(1), 105–107 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hasebe, T., Tsujie, S.: Order quasisymmetric functions distinguish rooted trees. J. Algebraic Combin. available online (2017)

  9. Jing-Ho, Y., Jer-Jeong, C., Chang, G.J.: Quasi-threshold graphs. Discrete Appl. Math.matics 69(3), 247–255 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd ed., Oxford mathematical monographs, 2nd edn. Clarendon Press; Oxford University Press, Oxford; New York (1995)

    Google Scholar 

  11. Read, R.C.: An introduction to chromatic polynomials. J. Combin. Theory 4(1), 52–71 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  12. Stanley, R.P., Stembridge, J.R.: On immanants of Jacobi–Trudi matrices and permutations with restricted position. J. Combin. Theory Ser. A 62(2), 261–279 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Stanley, R.P.: A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math. 111(1), 166–194 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Stanley, R.P.: Graph colorings and related symmetric functions: ideas and applications A description of results, interesting applications, and notable open problems. Discrete Math. 193(1), 267–286 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  16. Wolk, E.S.: The comparability graph of a tree. Proc. Am. Math. Soc. 13(5), 789–795 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wolk, E.S.: A note on “The comparability graph of a tree”. Proc. Am. Math. Soc. 16(1), 17–20 (1965)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to appreciate the anonymous referee for his/her valuable comments and careful reading.

Author information

Authors and Affiliations

  1. Department of Information Design, Hiroshima Kokusai Gakuin University, 6-20-1 Nakano, Aki-ku, Hiroshima, 739-0321, Japan

    Shuhei Tsujie

Authors
  1. Shuhei Tsujie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuhei Tsujie.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsujie, S. The Chromatic Symmetric Functions of Trivially Perfect Graphs and Cographs. Graphs and Combinatorics 34, 1037–1048 (2018). https://doi.org/10.1007/s00373-018-1928-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-018-1928-2

Keywords

Mathematics Subject Classification

Search

Navigation