Discrete Mathematics and Combinatorics Commons^™

Cohen-Macaulay Properties Of Closed Neighborhood Ideals, Jackson Leaman

All Theses

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 …

Matroid Generalizations Of Some Graph Results, Cameron Crenshaw

LSU Doctoral Dissertations

The edges of a graph have natural cyclic orderings. We investigate the matroids for which a similar cyclic ordering of the circuits is possible. A full characterization of the non-binary matroids with this property is given. Evidence of the difficulty of this problem for binary matroids is presented, along with a partial result for binary orderable matroids.

For a graph G, the ratio of |E(G)| to the minimum degree of G has a natural lower bound. For a matroid M that is representable over a finite field, we generalize this to a lower bound on …

Opposite Trees, Theo Goossens

Theses and Dissertations (Comprehensive)

A spanning tree of a graph G is a connected acyclic subgraph of G that includes all of the vertices in G. The degree of a vertex is the number of edges incident to that vertex. Given a spanning tree T of a graph G, an opposite tree of T is a spanning tree of G where the degree of each of its vertices is different from its degree in T. For complete, complete bipartite, and complete multipartite graphs, we give the conditions spanning trees of these graphs must satisfy in order to have an opposite tree.