Skip to main content Accessibility help
×
Home

Galois groups of chromatic polynomials

  • Kerri Morgan (a1)

Abstract

The chromatic polynomial P(G,λ) gives the number of ways a graph G can be properly coloured in at most λ colours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. We give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomials of non-clique-separable θ-graphs of order at most 19. Most of these chromatic polynomials have symmetric Galois groups. We give five infinite families of graphs: one of these families has chromatic polynomials with a dihedral Galois group and two of these families have chromatic polynomials with cyclic Galois groups. This includes the first known infinite family of graphs that have chromatic polynomials with the cyclic Galois group of order 3.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Galois groups of chromatic polynomials
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Galois groups of chromatic polynomials
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Galois groups of chromatic polynomials
      Available formats
      ×

Copyright

References

Hide All
[1]Beaudin, L., Ellis-Monaghan, J., Pangborn, G. and Shrock, R., ‘A little statistical mechanics for the graph theorist’, Discrete Math. 310 (2010) 20372053.
[2]Beraha, S., Unpublished, circa 1974.
[3]Beraha, S., Kahane, J. and Weiss, N. J., ‘Limits of zeros of recursively defined families of polynomials’, Studies in foundations and combinatorics: advances in mathematics supplementary studies, vol. 1 (ed. Rota, G.; Academic Press, New York, 1978) 213232.
[4]Beraha, S., Kahane, J. and Weiss, N. J., ‘Limits of chromatic zeros of some families of maps’, J. Combin. Theory Ser. B 28 (1980) 5265.
[5]Biggs, N. L., Damerell, R. M. and Sands, D. A., ‘Recursive families of graphs’, J. Combin. Theory Ser. B 12 (1972) 123131.
[6]Biggs, N. and Shrock, R., ‘T=0 partition functions for Potts antiferromagnets on square lattice strips with (twisted) periodic boundary conditions’, J. Phys. A: Math. Gen. 32 (1999) L489L493.
[7]Birkhoff, G. H., ‘A determinant formula for the number of ways of coloring a map’, Ann. of Math. 14 (1912–1913) 4246.
[8]Bohn, A., Cameron, P. J. and Müller, P., ‘Galois groups of multivariate Tutte polynomials’, available from http://arxiv.org/abs/1006.3869v2 [accessed June, 2011].
[9]Brookfield, G., ‘Factoring quartic polynomials: a lost art’, Math. Mag. 80 (2007) 6770.
[10]Cameron, P. and Morgan, K., ‘Algebraic properties of chromatic roots’, Submitted, 2011.
[11]Chang, S. and Shrock, R., ‘Ground-state entropy of the Potts antiferromagnet with next-nearest-neighbour spin-spin-couplings on strips of the square lattice’, Phys. Rev. E 62 (2000) 46504664.
[12]Conway, J. H., Hulpke, A. and McKay, J., ‘On transitive permutation groups’, LMS J. Comput. Math. 1 (1998) 18.
[13]Cox, D. A., Galois theory (Wiley-Interscience, New Jersey, 2004).
[14]D’Antonia, O. M., Mereghetti, C. and Zamparini, F., ‘The 224 non-chordal graphs on less than 10 vertices whose chromatic polynomials have no complex roots’, Discrete Math. 226 (2001) 387396.
[15]Farrell, E. J., ‘Chromatic roots—some observations and conjectures’, Discrete Math. 29 (1980) 161167.
[16]Gallian, J. A., Contemporary abstract algebra, 3rd edn (D.C. Heath and Company, Toronto, 1994).
[17]Jackson, B., ‘Zeros of chromatic and flow polynomials of graphs’, J. Geom. 76 (2003) 95109.
[18]Jacobsen, J. L. and Salas, J., ‘Transfer matrices and partition-function zeros for antiferromagnetic Potts models. II. Extended results for square-lattice chromatic polynomial’, J. Stat. Phys. 104 (2001) 701723.
[19]Jacobsen, J. L., Salas, J. and Sokal, A. D., ‘Transfer matrices and partition-function zeros for antiferromagnetic Potts models. III. Triangular-lattice chromatic polynomial’, J. Stat. Phys. 112 (2003) 9211017.
[20]McKay, B., ‘Simple graphs (connected)’, available from http://cs.anu.edu.au/people/bdm/data/graphs.html [accessed July 2006].
[21]Morgan, K., ‘Galois groups of chromatic polynomials of strongly-non-clique-separable graphs of order at most 10’, Technical Report 2009/234, 2009.
[22]Morgan, K., ‘Algebraic aspects of the chromatic polynomial’, PhD Thesis, Monash University, 2010, available from http://arrow.monash.edu.au/hdl/1959.1/470667.
[23]Morgan, K., ‘Pairs of chromatically equivalent graphs’, Graphs Combin. 27 (2011) 547556.
[24]Morgan, K. and Farr, G., ‘Certificates of factorisation for a class of triangle-free graphs’, Electron. J. Combin. 16 (2009). Research Paper R75.
[25]Morgan, K. and Farr, G., ‘Certificates of factorisation for chromatic polynomials’, Electron. J. Combin. 16 (2009). Research Paper R74.
[26]Morgan, K. and Farr, G., ‘Non-bipartite chromatic factors’, Discrete Math. 312 (2012) 11661170.
[27] ‘PARI/GP, version 2.3.0’, 2000, available from http://pari.math.u-bordeaux.fr/ [accessed February, 2006].
[28]Read, R. C., ‘An introduction to chromatic polynomials’, J. Combin. Theory 4 (1968) 5271.
[29]Read, R. C. and Royle, G. F., ‘Chromatic roots of families of graphs’, Graph theory, combinatorics and applications, Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, Kalmazoo, MI, 1988, vol. 2 (Wiley-Interscience, New York, 1991) 10091029.
[30]Read, R. C. and Tutte, W. T., ‘Chromatic polynomials’, Selected topics in graph theory, vol. 3 (eds Beineke, L. W. and Wilson, R. J.; Academic Press, London, 1988) 1542.
[31]Salas, J. and Sokal, A. D., ‘Transfer matrices and partition-function zeros for antiferromagnetic Potts models. I. General theory and square-lattice chromatic polynomial’, J. Stat. Phys. 104 (2001) 609699.
[32]Shrock, R., ‘T=0 partition functions for Potts antiferromagnets on Möbius strips and effects of graph topology’, Phys. Lett. A 261 (1999) 5762.
[33]Shrock, R. and Tsai, S., ‘Asymptotic limits and zeros of chromatic polynomials and ground state entropy of Potts antiferromagnets’, Phys. Rev. E 55 (1997) 51655178.
[34]Shrock, R. and Tsai, S., ‘Families of graphs with chromatic zeros lying in circles’, Phys. Rev. E 56 (1997) 13421345.
[35]Shrock, R. and Tsai, S., ‘Ground state entropy in Potts antiferromagnets and the approach to the two-dimensional thermodynamic limit’, Phys. Rev. E 58 (1998) 43324339.
[36]Shrock, R. and Tsai, S., ‘Ground state degeneracy of Potts antiferromagnets on two-dimensional lattices: approach using infinite cyclic strip graphs’, Phys. Rev. E 60 (1999) 35123515.
[37]Shrock, R. and Tsai, S., ‘Ground state entropy in Potts antiferromagnets and chromatic polynomials’, Nuclear Phys. B Proc. Suppl. 73 (1999) 751753.
[38]Shrock, R. and Tsai, S., ‘Ground-state entropy of the Potts antiferromagnet on cyclic strip graphs’, J. Phys. A 32 (1999) L195L200.
[39]Sokal, A. D., ‘Chromatic polynomials, Potts models and all that’, Phys. A 279 (2000) 324332.
[40]Sokal, A. D., ‘Bounds on the complex zeros of (di)chromatic polynomials and the Potts-model partition functions’, Combin. Probab. Comput. 10 (2001) 4177.
[41]Sokal, A. D., ‘Chromatic roots are dense in the whole complex plane’, Combin. Probab. Comput. 13 (2004) 221261.
[42]Tutte, W. T., ‘Chromials’, Hypergraph seminar, Lecture Notes in Mathematics 411 (eds Berge, C. and Ray-Chaudhuri, D.; Springer, Berlin, 1972) 243266.
[43]Tutte, W. T., ‘Chromatic sums for rooted planar triangulations: The cases λ=1 and λ=2’, Canad. J. Math. 25 (1974) 426447.
[44]Wieb, B., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) 235265.
[45]Woodall, D. R., ‘Zeros of chromatic polynomials’, Combinatorial Surveys: Proceedings of the Sixth British Combinatorial Conference (ed. Cameron, P. J.; Academic Press, London, 1977) 199223.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Related content

Powered by UNSILO

Galois groups of chromatic polynomials

  • Kerri Morgan (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.