210 research outputs found
Sinks in Acyclic Orientations of Graphs
Greene and Zaslavsky proved that the number of acyclic orientations of a
graph with a unique sink is, up to sign, the linear coefficient of the
chromatic polynomial. We give three new proofs of this result using pure
induction, noncommutative symmetric functions, and an algorithmic bijection.Comment: 17 pages, 1 figur
Listing Acyclic Orientations of Graphs with Single and Multiple Sources
International audienceWe study enumeration problems for the acyclic orientations of an undirected graph with n nodes and m edges, where each edge must be assigned a direction so that the resulting directed graph is acyclic. When the acyclic orientations have single or multiple sources specified as input along with the graph, our algorithm is the first one to provide guaranteed bounds, giving new bounds with a delay of O(mâ‹…n) time per solution and O(n2) working space. When no sources are specified, our algorithm improves over previous work by reducing the delay to O(m), and is the first one with linear delay
Set maps, umbral calculus, and the chromatic polynomial
Some important properties of the chromatic polynomial also hold for any
polynomial set map satisfying p_S(x+y)=\sum_{T\uplus U=S}p_T(x)p_U(y). Using
umbral calculus, we give a formula for the expansion of such a set map in terms
of any polynomial sequence of binomial type. This leads to some new expansions
of the chromatic polynomial. We also describe a set map generalization of Abel
polynomials.Comment: 20 page
Acyclic orientations on the Sierpinski gasket
We study the number of acyclic orientations on the generalized
two-dimensional Sierpinski gasket at stage with equal to
two and three, and determine the asymptotic behaviors. We also derive upper
bounds for the asymptotic growth constants for and -dimensional
Sierpinski gasket .Comment: 20 pages, 8 figures and 6 table
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
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