Intersections of Graph Theory and Combinatorial Commutative Algebra

Abstract

In this dissertation, we explore three problems in the fields of graph theory and combinatorial commutative algebra, and their intersections. First, we prove a higher upper bound for the slow coloring number of a certain class of graphs. Then we provide a novel construction of an old theorem characterizing well-covered bipartite graphs, along with a new proof. Our proof gives more insight into the structure of this class of graphs. Finally, we generate a class of bipartite graphs whose edge ideals are Bi-Cohen-Macaulay, a very strong property that endows these graphs with rich algebraic structure. This work demonstrates the usefulness of viewing the same mathematical