arXiv:2611.13273v1 Announce Type: new Abstract: The chromatic polynomial is a well-studied function of graphs whose coefficients, factorization, and other properties are well-known to encode structural information about the underlying graph. An analogous notion of graph colouring exists for directed graphs which is also well-studied, but little is yet known about the corresponding dichromatic polynomial. We investigate properties of the dichromatic polynomial, including its coefficient interpretations, root-dense regions, and asymptotic growth. We also show that chordal digraphs' polynomials split into linear factors in a way analogous to the undirected case, confirm a conjecture on dichromatic uniqueness posed by Gonz\'alez-Moreno et al., and study the density of dichromatic polynomials within certain larger families of polynomials.
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