45,825 research outputs found
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
Powers of Hamilton cycles in pseudorandom graphs
We study the appearance of powers of Hamilton cycles in pseudorandom graphs,
using the following comparatively weak pseudorandomness notion. A graph is
-pseudorandom if for all disjoint and with and we have
. We prove that for all there is an
such that an -pseudorandom graph on
vertices with minimum degree at least contains the square of a
Hamilton cycle. In particular, this implies that -graphs with
contain the square of a Hamilton cycle, and thus
a triangle factor if is a multiple of . This improves on a result of
Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random
graphs, Combinatorica 24 (2004), no. 3, 403--426].
We also extend our result to higher powers of Hamilton cycles and establish
corresponding counting versions.Comment: 30 pages, 1 figur
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
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