314 research outputs found
Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero
(with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27].
Analogous theorems hold for the flow polynomial of bridgeless graphs and for
the characteristic polynomial of loopless matroids. Here we exhibit all these
results as special cases of more general theorems on real zero-free regions of
the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and
employ deletion-contraction together with parallel and series reduction. In
particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure
The Tutte dichromate and Whitney homology of matroids
We consider a specialization of the Tutte polynomial of a matroid
which is inspired by analogy with the Potts model from statistical
mechanics. The only information lost in this specialization is the number of
loops of . We show that the coefficients of are very simply
related to the ranks of the Whitney homology groups of the opposite partial
orders of the independent set complexes of the duals of the truncations of .
In particular, we obtain a new homological interpretation for the coefficients
of the characteristic polynomial of a matroid
Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements
Hyperplanes of the form x_j = x_i + c are called affinographic. For an
affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we
study the function f(M) that counts integral points in [1,M]^n that do not lie
in any hyperplane of the arrangement. We show that f(M) is a piecewise
polynomial function of positive integers M, composed of terms that appear
gradually as M increases. Our approach is to convert the problem to one of
counting integral proper colorations of a rooted integral gain graph. An
application is to interval coloring in which the interval of available colors
for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo
M; the number of proper modular colorations is a different piecewise polynomial
that for large M becomes the characteristic polynomial of the arrangement (by
which means Athanasiadis previously obtained that polynomial). We also study
this function for all positive moduli.Comment: 13 p
Coloring Complexes and Combinatorial Hopf Monoids
We generalize the notion of coloring complex of a graph to linearized
combinatorial Hopf monoids. These are a generalization of the notion of
coloring complex of a graph. We determine when a combinatorial Hopf monoid has
such a construction, and discover some inequalities that are satisfied by the
quasisymmetric function invariants associated to the combinatorial Hopf monoid.
We show that the collection of all such coloring complexes forms a
combinatorial Hopf monoid, which is the terminal object in the category of
combinatorial Hopf monoids with convex characters. We also study several
examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure
A unique factorization theorem for matroids
We study the combinatorial, algebraic and geometric properties of the free
product operation on matroids. After giving cryptomorphic definitions of free
product in terms of independent sets, bases, circuits, closure, flats and rank
function, we show that free product, which is a noncommutative operation, is
associative and respects matroid duality. The free product of matroids and
is maximal with respect to the weak order among matroids having as a
submatroid, with complementary contraction equal to . Any minor of the free
product of and is a free product of a repeated truncation of the
corresponding minor of with a repeated Higgs lift of the corresponding
minor of . We characterize, in terms of their cyclic flats, matroids that
are irreducible with respect to free product, and prove that the factorization
of a matroid into a free product of irreducibles is unique up to isomorphism.
We use these results to determine, for K a field of characteristic zero, the
structure of the minor coalgebra of a family of matroids that
is closed under formation of minors and free products: namely, is
cofree, cogenerated by the set of irreducible matroids belonging to .Comment: Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for
publication in the Journal of Combinatorial Theory (A). See
arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this
subjec
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