Research Problems from the BCC21

AbstractA collection of open problems, mostly presented at the problem session of the 21st British Combinatorial Conference

Chromatic Bounds on Orbital Chromatic Roots

Given a group G of automorphisms of a graph Γ, the orbital chromatic polynomial OPΓ,G(x) is the polynomial whose value at a positive integer k is the number of orbits of G on proper k-colorings of Γ. In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in R, extending Thomassen\u27s famous result (see \cite{Thomassen}) that chromatic roots are dense in [32/27,∞). Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root of its chromatic polynomial. We resolve this conjecture in the negative, and provide a process for generating families of counterexamples. We additionally show that the answer is true for various classes of graphs, including many outerplanar graphs

Chromatic Quasisymmetric Class Functions for combinatorial Hopf monoids

We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid H, and an H-structure h on a set N, there are proper colorings of h, generalizing graph colorings and poset partitions. We show that the automorphism group of h acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley\u27s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We also introduce the flag quasisymmetric class function of a balanced relative simplicial complex equipped with a group action. We show that, under certain conditions, the chromatic quasisymmetric class function of h is the flag quasisymmetric class function of a balanced relative simplicial complex that we call the coloring complex of h. We use this result to deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others

The Chromatic Quasisymmetric Class Function of a Digraph

We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the former is a generalization of a quasisymmetric function introduced by Grinberg. We prove representation-theoretic analogues of classical and recent results, including F-positivity, and combinatorial reciprocity theorems. We deduce results for orbital quasisymmetric functions, and study a generalization of the notion of strongly flawless sequences

Combinatorial Hopf algebras from representations of families of wreath products

We construct Hopf algebras whose elements are representations of combinatorial automorphism groups, by generalising a theorem of Zelevinsky on Hopf algebras of representations of wreath products. As an application we attach symmetric functions to representations of graph automorphism groups, generalising and refining Stanley's chromatic symmetric function.Comment: 26 page

Orbital chromatic and flow roots

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 [57,00), Here we discuss the location of real orbital chromatic roots. We show, for example, that they are dense in ℝ, 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.</p