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Galois groups of chromatic polynomials

Part of: Algebraic number theory: global fields Graph theory

Published online by Cambridge University Press:  01 September 2012

Kerri Morgan
Kerri Morgan*
Affiliation:
Clayton School of Information Technology, Monash University, Building 63, Wellington Road, Clayton 3800, Australia (email: Kerri.Morgan@monash.edu)

Abstract

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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.

MSC classification

Secondary: 05C15: Coloring of graphs and hypergraphs 11R32: Galois theory 05C31: Graph polynomials
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 281 - 307
Copyright
Copyright © London Mathematical Society 2012

References

[1]Beaudin, L., Ellis-Monaghan, J., Pangborn, G. and Shrock, R., ‘A little statistical mechanics for the graph theorist’, Discrete Math. 310 (2010) 2037–2053.CrossRefGoogle Scholar
[2]Beraha, S., Unpublished, circa 1974.Google Scholar
[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) 213–232.Google Scholar
[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) 52–65.CrossRefGoogle Scholar
[5]Biggs, N. L., Damerell, R. M. and Sands, D. A., ‘Recursive families of graphs’, J. Combin. Theory Ser. B 12 (1972) 123–131.CrossRefGoogle Scholar
[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) L489–L493.CrossRefGoogle Scholar
[7]Birkhoff, G. H., ‘A determinant formula for the number of ways of coloring a map’, Ann. of Math. 14 (1912–1913) 42–46.CrossRefGoogle Scholar
[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].CrossRefGoogle Scholar
[9]Brookfield, G., ‘Factoring quartic polynomials: a lost art’, Math. Mag. 80 (2007) 67–70.CrossRefGoogle Scholar
[10]Cameron, P. and Morgan, K., ‘Algebraic properties of chromatic roots’, Submitted, 2011.Google Scholar
[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) 4650–4664.CrossRefGoogle Scholar
[12]Conway, J. H., Hulpke, A. and McKay, J., ‘On transitive permutation groups’, LMS J. Comput. Math. 1 (1998) 1–8.CrossRefGoogle Scholar
[13]Cox, D. A., Galois theory (Wiley-Interscience, New Jersey, 2004).CrossRefGoogle Scholar
[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) 387–396.CrossRefGoogle Scholar
[15]Farrell, E. J., ‘Chromatic roots—some observations and conjectures’, Discrete Math. 29 (1980) 161–167.CrossRefGoogle Scholar
[16]Gallian, J. A., Contemporary abstract algebra, 3rd edn (D.C. Heath and Company, Toronto, 1994).Google Scholar
[17]Jackson, B., ‘Zeros of chromatic and flow polynomials of graphs’, J. Geom. 76 (2003) 95–109.CrossRefGoogle Scholar
[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) 701–723.CrossRefGoogle Scholar
[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) 921–1017.CrossRefGoogle Scholar
[20]McKay, B., ‘Simple graphs (connected)’, available from http://cs.anu.edu.au/people/bdm/data/graphs.html [accessed July 2006].Google Scholar
[21]Morgan, K., ‘Galois groups of chromatic polynomials of strongly-non-clique-separable graphs of order at most 10’, Technical Report 2009/234, 2009.Google Scholar
[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.Google Scholar
[23]Morgan, K., ‘Pairs of chromatically equivalent graphs’, Graphs Combin. 27 (2011) 547–556.CrossRefGoogle Scholar
[24]Morgan, K. and Farr, G., ‘Certificates of factorisation for a class of triangle-free graphs’, Electron. J. Combin. 16 (2009). Research Paper R75.CrossRefGoogle Scholar
[25]Morgan, K. and Farr, G., ‘Certificates of factorisation for chromatic polynomials’, Electron. J. Combin. 16 (2009). Research Paper R74.CrossRefGoogle Scholar
[26]Morgan, K. and Farr, G., ‘Non-bipartite chromatic factors’, Discrete Math. 312 (2012) 1166–1170.CrossRefGoogle Scholar
[27] ‘PARI/GP, version 2.3.0’, 2000, available from http://pari.math.u-bordeaux.fr/ [accessed February, 2006].Google Scholar
[28]Read, R. C., ‘An introduction to chromatic polynomials’, J. Combin. Theory 4 (1968) 52–71.CrossRefGoogle Scholar
[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) 1009–1029.Google Scholar
[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) 15–42.Google Scholar
[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) 609–699.CrossRefGoogle Scholar
[32]Shrock, R., ‘T=0 partition functions for Potts antiferromagnets on Möbius strips and effects of graph topology’, Phys. Lett. A 261 (1999) 57–62.CrossRefGoogle Scholar
[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) 5165–5178.CrossRefGoogle Scholar
[34]Shrock, R. and Tsai, S., ‘Families of graphs with chromatic zeros lying in circles’, Phys. Rev. E 56 (1997) 1342–1345.CrossRefGoogle Scholar
[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) 4332–4339.CrossRefGoogle Scholar
[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) 3512–3515.CrossRefGoogle ScholarPubMed
[37]Shrock, R. and Tsai, S., ‘Ground state entropy in Potts antiferromagnets and chromatic polynomials’, Nuclear Phys. B Proc. Suppl. 73 (1999) 751–753.CrossRefGoogle Scholar
[38]Shrock, R. and Tsai, S., ‘Ground-state entropy of the Potts antiferromagnet on cyclic strip graphs’, J. Phys. A 32 (1999) L195–L200.CrossRefGoogle Scholar
[39]Sokal, A. D., ‘Chromatic polynomials, Potts models and all that’, Phys. A 279 (2000) 324–332.CrossRefGoogle Scholar
[40]Sokal, A. D., ‘Bounds on the complex zeros of (di)chromatic polynomials and the Potts-model partition functions’, Combin. Probab. Comput. 10 (2001) 41–77.CrossRefGoogle Scholar
[41]Sokal, A. D., ‘Chromatic roots are dense in the whole complex plane’, Combin. Probab. Comput. 13 (2004) 221–261.CrossRefGoogle Scholar
[42]Tutte, W. T., ‘Chromials’, Hypergraph seminar, Lecture Notes in Mathematics 411 (eds Berge, C. and Ray-Chaudhuri, D.; Springer, Berlin, 1972) 243–266.Google Scholar
[43]Tutte, W. T., ‘Chromatic sums for rooted planar triangulations: The cases λ=1 and λ=2’, Canad. J. Math. 25 (1974) 426–447.CrossRefGoogle Scholar
[44]Wieb, B., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) 235–265.Google Scholar
[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) 199–223.Google Scholar