524 research outputs found
Bipartite powers of k-chordal graphs
Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have
an induced cycle of length greater than k. From the definition it is clear that
3-chordal graphs are precisely the class of chordal graphs. Duchet proved that,
for every positive integer m, if G^m is chordal then so is G^{m+2}.
Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak.
Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics,
177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}.
Powering a bipartite graph does not preserve its bipartitedness. In order to
preserve the bipartitedness of a bipartite graph while powering Chandran et al.
introduced the notion of bipartite powering. This notion was introduced to aid
their study of boxicity of chordal bipartite graphs. Given a bipartite graph G
and an odd positive integer m, we define the graph G^{[m]} to be a bipartite
graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is
odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power
of G.
In this paper we show that, given a bipartite graph G, if G is k-chordal then
so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is
odd.Comment: 10 page
Ideals with componentwise linear powers
Let be the polynomial ring over a field , and let
be a finitely generated standard graded -algebra. We show that if the
defining ideal of has a quadratic initial ideal, then all the graded
components of are componentwise linear. Applying this result to the Rees
ring of a graded ideal gives a criterion on to have
componentwise linear powers. Moreover, for any given graph , a construction
on is presented which produces graphs whose cover ideals have
componentwise linear powers. This in particular implies that for any
Cohen-Macaulay Cameron-Walker graph all powers of have linear
resolutions. Moreover, forming a cone on special graphs like unmixed chordal
graphs, path graphs and Cohen-Macaulay bipartite graphs produces cover ideals
with componentwise linear powers
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
Regularity of squarefree monomial ideals
We survey a number of recent studies of the Castelnuovo-Mumford regularity of
squarefree monomial ideals. Our focus is on bounds and exact values for the
regularity in terms of combinatorial data from associated simplicial complexes
and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.
- …