580 research outputs found
On the commutative quotient of Fomin-Kirillov algebras
The Fomin-Kirillov algebra is a noncommutative algebra with a
generator for each edge in the complete graph on vertices. For any graph
on vertices, let be the subalgebra of
generated by the edges in . We show that the commutative quotient of
is isomorphic to the Orlik-Terao algebra of . As a
consequence, the Hilbert series of this quotient is given by , where is the chromatic polynomial of . We also
give a reduction algorithm for the graded components of that do
not vanish in the commutative quotient and show that their structure is
described by the combinatorics of noncrossing forests.Comment: 11 pages, 3 figure
Subword complexes via triangulations of root polytopes
Subword complexes are simplicial complexes introduced by Knutson and Miller
to illustrate the combinatorics of Schubert polynomials and determinantal
ideals. They proved that any subword complex is homeomorphic to a ball or a
sphere and asked about their geometric realizations. We show that a family of
subword complexes can be realized geometrically via regular triangulations of
root polytopes. This implies that a family of -Grothendieck polynomials
are special cases of reduced forms in the subdivision algebra of root
polytopes. We can also write the volume and Ehrhart series of root polytopes in
terms of -Grothendieck polynomials.Comment: 17 pages, 15 figure
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
Flow polytopes of signed graphs and the Kostant partition function
We establish the relationship between volumes of flow polytopes associated to
signed graphs and the Kostant partition function. A special case of this
relationship, namely, when the graphs are signless, has been studied in detail
by Baldoni and Vergne using techniques of residues. In contrast with their
approach, we provide entirely combinatorial proofs inspired by the work of
Postnikov and Stanley on flow polytopes. As a fascinating special family of
flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the
beautiful volume formula for the type version,
where is the th Catalan number, we introduce type and
Chan-Robbins-Yuen polytopes along with intriguing conjectures
pertaining to their properties.Comment: 29 pages, 13 figure
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