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Orbital Chromatic and Flow Roots

  • PETER J. CAMERON (a1) and K. K. KAYIBI (a1)

The chromatic polynomial PΓ(x) of a graph Γ is a polynomial whose value at the positive integer k is the number of proper k-colourings of Γ. If G is a group of automorphisms of Γ, then there is a polynomial OPΓ,G(x), whose value at the positive integer k is the number of orbits of G on proper k-colourings of Γ.

It is known that real chromatic roots cannot be negative, but they are dense in ∞). Here we discuss the location of real orbital chromatic roots. We show, for example, that they are dense in , but under certain hypotheses, there are zero-free regions.

We also look at orbital flow roots. Here things are more complicated because the orbit count is given by a multivariate polynomial; but it has a natural univariate specialization, and we show that the roots of these polynomials are dense in the negative real axis.

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[2] B. Jackson (1993) A zero-free interval for chromatic polynomials of graphs. Combin. Probab. Comput. 2 325336.

[4] A. D. Sokal (2004) Chromatic roots are dense in the whole complex plane. Combin. Probab. Comput. 13 221261.

[5] C. Thomassen (1997) The zero-free intervals for chromatic polynomials of graphs. Combin. Probab. Comput. 6 497506.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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