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Buchstaber numbers and classical invariants of simplicial complexes
Buchstaber invariant is a numerical characteristic of a simplicial complex,
arising from torus actions on moment-angle complexes. In the paper we study the
relation between Buchstaber invariants and classical invariants of simplicial
complexes such as bigraded Betti numbers and chromatic invariants. The
following two statements are proved. (1) There exists a simplicial complex U
with different real and ordinary Buchstaber invariants. (2) There exist two
simplicial complexes with equal bigraded Betti numbers and chromatic numbers,
but different Buchstaber invariants. To prove the first theorem we define
Buchstaber number as a generalized chromatic invariant. This approach allows to
guess the required example. The task then reduces to a finite enumeration of
possibilities which was done using GAP computational system. To prove the
second statement we use properties of Taylor resolutions of face rings.Comment: 19 pages, 2 figure
Boolean complexes for Ferrers graphs
In this paper we provide an explicit formula for calculating the boolean
number of a Ferrers graph. By previous work of the last two authors, this
determines the homotopy type of the boolean complex of the graph. Specializing
to staircase shapes, we show that the boolean numbers of the associated Ferrers
graphs are the Genocchi numbers of the second kind, and obtain a relation
between the Legendre-Stirling numbers and the Genocchi numbers of the second
kind. In another application, we compute the boolean number of a complete
bipartite graph, corresponding to a rectangular Ferrers shape, which is
expressed in terms of the Stirling numbers of the second kind. Finally, we
analyze the complexity of calculating the boolean number of a Ferrers graph
using these results and show that it is a significant improvement over
calculating by edge recursion.Comment: final version, to appear in the The Australasian Journal of
Combinatoric
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
The Coloring Ideal and Coloring Complex of a Graph
Let be a simple graph on vertices. We define a monomial ideal in
the Stanley-Reisner ring of the order complex of the Boolean algebra on
atoms. The monomials in are in one-to-one correspondence with the proper
colorings of . In particular, the Hilbert polynomial of equals the
chromatic polynomial of .
The ideal is generated by square-free monomials, so is the
Stanley-Reisner ring of a simplicial complex . The -vector of is a
certain transformation of the tail of the chromatic polynomial
of . The combinatorial structure of the complex is described
explicitly and it is shown that the Euler characteristic of equals the
number of acyclic orientations of .Comment: 13 pages, 3 figure
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