344 research outputs found
A conjecture on critical graphs and connections to the persistence of associated primes
We introduce a conjecture about constructing critically (s+1)-chromatic
graphs from critically s-chromatic graphs. We then show how this conjecture
implies that any unmixed height two square-free monomial ideal I, i.e., the
cover ideal of a finite simple graph, has the persistence property, that is,
Ass(R/I^s) \subseteq Ass(R/I^{s+1}) for all s >= 1. To support our conjecture,
we prove that the statement is true if we also assume that \chi_f(G), the
fractional chromatic number of the graph G, satisfies \chi(G) -1 < \chi_f(G) <=
\chi(G). We give an algebraic proof of this result.Comment: 11 pages; Minor changes throughout the paper; to appear in Discrete
Math
Replication in critical graphs and the persistence of monomial ideals
Motivated by questions about square-free monomial ideals in polynomial rings,
in 2010 Francisco et al. conjectured that for every positive integer k and
every k-critical (i.e., critically k-chromatic) graph, there is a set of
vertices whose replication produces a (k+1)-critical graph. (The replication of
a set W of vertices of a graph is the operation that adds a copy of each vertex
w in W, one at a time, and connects it to w and all its neighbours.)
We disprove the conjecture by providing an infinite family of
counterexamples. Furthermore, the smallest member of the family answers a
question of Herzog and Hibi concerning the depth functions of square-free
monomial ideals in polynomial rings, and a related question on the persistence
property of such ideals
Persistence and stability properties of powers of ideals
We introduce the concept of strong persistence and show that it implies
persistence regarding the associated prime ideals of the powers of an ideal. We
also show that strong persistence is equivalent to a condition on power of
ideals studied by Ratliff. Furthermore, we give an upper bound for the depth of
powers of monomial ideals in terms of their linear relation graph, and apply
this to show that the index of depth stability and the index of stability for
the associated prime ideals of polymatroidal ideals is bounded by their
analytic spread.Comment: 15 pages, 1 figur
Generalized cover ideals and the persistence property
Let be a square-free monomial ideal in , and
consider the sets of associated primes for all integers . Although it is known that the sets of associated primes of powers of
eventually stabilize, there are few results about the power at which this
stabilization occurs (known as the index of stability). We introduce a family
of square-free monomial ideals that can be associated to a finite simple graph
that generalizes the cover ideal construction. When is a tree, we
explicitly determine for all . As consequences, not
only can we compute the index of stability, we can also show that this family
of ideals has the persistence property.Comment: 15 pages; revised version has a new introduction; references updated;
to appear in J. Pure. Appl. Algebr
The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an
ordered proper vertex colouring of the complex. This ideal is a monomial ideal
which is generally not squarefree. We show that such a monomial ideal has a
linear resolution, as do all of its powers, if and only if the colouring
satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular
resolution supported on a cubical complex. From this, we give the graded Betti
numbers in terms of the face-vector of the underlying simplicial complex.
Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both
the ideal and its quotient. We also give explicit formul\ae\ for the
codimension, Krull dimension, multiplicity, projective dimension, depth, and
regularity. Further still, we describe the associated primes, and we show that
they are persistent.Comment: 34 pages, 8 figure
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