355 research outputs found
Cohen-Macaulay Properties of Closed Neighborhood Ideals
This thesis investigates Cohen-Macaulay properties of squarefree monomial ideals, which is an important line of inquiry in the field of combinatorial commutative algebra. A famous example of this is Villareal’s edge ideal [11]: given a finite simple graph G with vertices x1, . . . , xn, the edge ideal of G is generated by all the monomials of the form xixj where xi and xj are adjacent in G. Villareal’s characterization of Cohen-Macaulay edge ideals associated to trees is an often-cited result in the literature. This was extended to chordal and bipartite graphs by Herzog, Hibi, and Zheng in [7] and by Herzog and Hibi in [6]. In 2020, Sharifan and Moradi [10] introduced a related construction called the closed neighborhood ideal of a graph. Whereas an edge ideal of a graph G is generated by monomials associated to each edge in G, the closed neighborhood ideal is generated by monomials associated to its closed neighborhoods. In 2021, Sather-Wagstaff and Honeycutt [8] characterized trees whose closed neighborhood ideals are Cohen-Macaulay. We will provide a generalization of this characterization to chordal graphs and bipartite graphs. Additionally, we will survey the behavior of the depth of closed neighborhood ideals under certain graph operations
Componentwise Linearity of Powers of Cover Ideals
Let be a finite simple graph and denote its vertex cover ideal in
a polynomial ring over a field. Assume that is its -th symbolic
power. In this paper, we give a criteria for cover ideals of vertex
decomposable graphs to have the property that all their symbolic powers are not
componentwise linear. Also, we give a necessary and sufficient condition on
so that is a componentwise linear ideal for some (equivalently,
for all) when is a graph such that has a
simplicial vertex for any independent set of . Using this result, we
prove that is a componentwise linear ideal for several classes of
graphs for all . In particular, if is a bipartite graph, then
is a componentwise linear ideal if and only if is a
componentwise linear ideal for some (equivalently, for all) .Comment: arXiv admin note: text overlap with arXiv:1908.1057
Completely Independent Spanning Trees in Some Regular Graphs
Let be an integer and be spanning trees of a graph
. If for any pair of vertices of , the paths from to
in each , , do not contain common edges and common vertices,
except the vertices and , then are completely
independent spanning trees in . For -regular graphs which are
-connected, such as the Cartesian product of a complete graph of order
and a cycle and some Cartesian products of three cycles (for ), the
maximum number of completely independent spanning trees contained in these
graphs is determined and it turns out that this maximum is not always
Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach
Chordal structure and bounded treewidth allow for efficient computation in
numerical linear algebra, graphical models, constraint satisfaction and many
other areas. In this paper, we begin the study of how to exploit chordal
structure in computational algebraic geometry, and in particular, for solving
polynomial systems. The structure of a system of polynomial equations can be
described in terms of a graph. By carefully exploiting the properties of this
graph (in particular, its chordal completions), more efficient algorithms can
be developed. To this end, we develop a new technique, which we refer to as
chordal elimination, that relies on elimination theory and Gr\"obner bases. By
maintaining graph structure throughout the process, chordal elimination can
outperform standard Gr\"obner basis algorithms in many cases. The reason is
that all computations are done on "smaller" rings, of size equal to the
treewidth of the graph. In particular, for a restricted class of ideals, the
computational complexity is linear in the number of variables. Chordal
structure arises in many relevant applications. We demonstrate the suitability
of our methods in examples from graph colorings, cryptography, sensor
localization and differential equations.Comment: 40 pages, 5 figure
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