120,352 research outputs found
The toric face ring of a simplicial complex
We consider standard graded toric rings whose generators
correspond to the faces of a simplicial complex . When is
normal, it is shown that its divisor class group is free. For a flag complex
which is the clique complex of a perfect graph, a nice description for
the class group and the canonical module of in terms of the
minimal vertex covers of the graph is given. Moreover, for a quasi-forest
simplicial complex a quadratic Gr\"obner basis for the defining ideal of
is presented. Using this fact we give combinatorial descriptions
for the -invariant and the Gorenstein property of .Comment: 15 page
Line-graphs of cubic graphs are normal
A graph is called normal if its vertex set can be covered by cliques and also
by stable sets, such that every such clique and stable set have non-empty
intersection. This notion is due to Korner, who introduced the class of normal
graphs as an extension of the class of perfect graphs. Normality has also
relevance in information theory. Here we prove, that the line graphs of cubic
graphs are normal.Comment: 16 pages, 10 figure
Ehrhart clutters: Regularity and Max-Flow Min-Cut
If C is a clutter with n vertices and q edges whose clutter matrix has column
vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a
Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to
show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then
C is an Ehrhart clutter and in this case we provide sharp bounds on the
Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols
conjecture on packing problems, we conjecture that if C is both ideal and the
clique clutter of a perfect graph, then C has the MFMC property. We prove this
conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel
graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof
of our conjecture when C is a uniform clique clutter of a perfect graph. We
close with a generalization of Ehrhart clutters as it relates to total dual
integrality.Comment: Electronic Journal of Combinatorics, to appea
Minimal covers of the prisms and antiprisms
This paper contains a classication of the regular minimal abstract polytopes
that act as covers for the convex polyhedral prisms and antiprisms. It includes
a detailed discussion of their topological structure, and completes the
enumeration of such covers for convex uniform polyhedra. Additionally, this
paper addresses related structural questions in the theory of string C-groups.Comment: 22 pages, 8 figure
On the dimension of the minimal vertex covers semigroup ring of an unmixed bipartite graph
In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the
semigroup ring associated to the set of minimal vertex covers of an unmixed
bipartite graph. In this paper we relate the dimension of this semigroup ring
to the rank of the Boolean lattice associated to the graph.Comment: 6 pages, Pragmatic 2008, University of Catania (Italy); corrected
typo
Universal abelian covers of superisolated singularities
We give explicit examples of Gorenstein surface singularities with integral
homology sphere link, which are not complete intersections. Their existence was
shown by Luengo-Velasco, Melle-Hernandez and Nemethi, thereby providing
counterexamples to the Universal abelian covering conjecture of Neumann and
Wahl.Comment: Some examples and explanations added; updated version. 23 page
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