6,844 research outputs found
Evaluations of topological Tutte polynomials
We find new properties of the topological transition polynomial of embedded
graphs, . We use these properties to explain the striking similarities
between certain evaluations of Bollob\'as and Riordan's ribbon graph
polynomial, , and the topological Penrose polynomial, . The general
framework provided by also leads to several other combinatorial
interpretations these polynomials. In particular, we express , ,
and the Tutte polynomial, , as sums of chromatic polynomials of graphs
derived from ; show that these polynomials count -valuations of medial
graphs; show that counts edge 3-colourings; and reformulate the Four
Colour Theorem in terms of . We conclude with a reduction formula for the
transition polynomial of the tensor product of two embedded graphs, showing
that it leads to additional relations among these polynomials and to further
combinatorial interpretations of and .Comment: V2: major revision, several new results, and improved expositio
A categorification of the chromatic symmetric polynomial
International audienceThe Stanley chromatic polynomial of a graph is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology *() of graded -modules, whose graded Frobenius series reduces to the chromatic symmetric function at . We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology.Le polynôme chromatique symétrique d’un graphe est une généralisation par une fonction symétrique du polynôme chromatique, et possède des propriétés combinatoires intéressantes. Nous appliquons les techniques de l’homologie de Khovanov pour construire une homologie *() de modules gradués , dont la série bigraduée de Frobeniusse réduit au polynôme chromatique symétrique à . Nous obtenons également des analogies pour plusieurs propriétés connues des polynômes chromatiques en termes d’homologie
A rooted variant of Stanley's chromatic symmetric function
Richard Stanley defined the chromatic symmetric function of a graph
and asked whether there are non-isomorphic trees and with . We
study variants of the chromatic symmetric function for rooted graphs, where we
require the root vertex to either use or avoid a specified color. We present
combinatorial identities and recursions satisfied by these rooted chromatic
polynomials, explain their relation to pointed chromatic functions and rooted
-polynomials, and prove three main theorems. First, for all non-empty
connected graphs , Stanley's polynomial is irreducible
in for all large enough . The same result holds
for our rooted variant where the root node must avoid a specified color. We
prove irreducibility by a novel combinatorial application of Eisenstein's
Criterion. Second, we prove the rooted version of Stanley's Conjecture: two
rooted trees are isomorphic as rooted graphs if and only if their rooted
chromatic polynomials are equal. In fact, we prove that a one-variable
specialization of the rooted chromatic polynomial (obtained by setting
, , and for ) already distinguishes rooted
trees. Third, we answer a question of Pawlowski by providing a combinatorial
interpretation of the monomial expansion of pointed chromatic functions.Comment: 21 pages; v2: added a short algebraic proof to Theorem 2 (now Theorem
15), we also answer a question of Pawlowski about monomial expansions; v3:
added additional one-variable specialization results, simplified main proof
Building fences around the chromatic coefficients.
Although these bounding conditions do not allow us to completely predict all chromatic polynomials, they do serve to severely limit the form of polynomials considered to be candidates for the chromatic polynomial of some graph.In general it remains an unsolved problem to determine which polynomials are chromatic. The purpose of this paper is to establish controls over the allowable values and patterns in the coefficients of chromatic polynomials.Associated to each graph G is its chromatic polynomial and we associate to the sequence of the norms of its coefficients. A stringent partial ordering is established for such sequences. First, we show that if H is a subgraph of G then The main result is that for any graph G with q edges we have \alpha (R\sb{q}) \le \alpha (G) \le (S\sb{q}), where R\sb{q} and S\sb{q} are specified graphs with q edges. It is also useful to examine the coefficient sequence of a chromatic polynomial which has been expressed in terms of falling factorials. If G has m missing edges we find that \beta (T\sb{\bar m})\le \beta (G)\le \beta (X\sb{\bar m}) where T\sb{\bar m} and X\sb{\bar m} are specified graphs with m missing edges
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