210 research outputs found
Notes on acyclic orientations and the shelling lemma
AbstractIn this paper we study two lemmas on acyclic orientations and totally cyclic orientations of a graph, which can be derived from the shelling lemma in vector subspaces. We give simple graph theoretical proofs as well as a proof by the interpretations of the shelling lemma in the special setting of graphs. Furthermore, we present similar interpretations of closely related theorems in vector subspaces, which do not seem to admit simple graph theoretical proofs
Some inequalities for the Tutte polynomial
We prove that the Tutte polynomial of a coloopless paving matroid is convex
along the portions of the line segments x+y=p lying in the positive quadrant.
Every coloopless paving matroids is in the class of matroids which contain two
disjoint bases or whose ground set is the union of two bases of M*. For this
latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a
>= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same
class of matroids. We also prove this conjecture for some families of graphs
and matroids.Comment: 17 page
Morphisms and order ideals of toric posets
Toric posets are cyclic analogues of finite posets. They can be viewed
combinatorially as equivalence classes of acyclic orientations generated by
converting sources into sinks, or geometrically as chambers of toric graphic
hyperplane arrangements. In this paper we study toric intervals, morphisms, and
order ideals, and we provide a connection to cyclic reducibility and conjugacy
in Coxeter groups.Comment: 28 pages, 8 figures. A 12-page "extended abstract" version appears as
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The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations
We show that the 4-variable generating function of certain orientation
related parameters of an ordered oriented matroid is the evaluation at (x + u,
y+v) of its Tutte polynomial. This evaluation contains as special cases the
counting of regions in hyperplane arrangements and of acyclic orientations in
graphs. Several new 2-variable expansions of the Tutte polynomial of an
oriented matroid follow as corollaries.
This result hold more generally for oriented matroid perspectives, with
specific special cases the counting of bounded regions in hyperplane
arrangements or of bipolar acyclic orientations in graphs.
In corollary, we obtain expressions for the partial derivatives of the Tutte
polynomial as generating functions of the same orientation parameters.Comment: 23 pages, 2 figures, 3 table
Toric partial orders
We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders
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