The Zero-Free Intervals for Chromatic Polynomials of Graphs
Combinatorics, Probability and Computing
Chromatic Roots are Dense in the Whole Complex Plane
Combinatorics, Probability and Computing
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The chromatic polynomial PΓ(x) of a graph“ is a polynomial whose value at the positive integerk is the number of proper k-colourings of Γ. IfG is a group of automorphisms of Γ, then there is apolynomial OPΓ,G(x), whose value atthe positive integer k is the number of orbits of Gon proper k-colourings of “.It is known that real chromatic roots cannot be negative, butthey are dense in [32/27·∞). Here we discuss thelocation of real orbital chromatic roots. We show, for example,that they are dense in ℝ, but under certain hypotheses, thereare zero-free regions.We also look at orbital flow roots. Here things are morecomplicated because the orbit count is given by a multivariatepolynomial; but it has a natural univariate specialization, and weshow that the roots of these polynomials are dense in the negativereal axis.