## Doctoral Dissertations

Spring 2011

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Computational Analysis and Modeling

Richard J. Greechie

#### Abstract

This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce "π-versions" of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π, if a ✶ (bc) ≤ (ab) ✶ c for all 3-element antichains { a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chains and define the order- skeleton of a lattice L to be := L/∼. We prove that the following are equivalent for a lattice L: (i) L satisfies D0π, ( ii) satisfies any of the five π-versions of distributivity, (iii) the order-skeleton is distributive.

Second, the symmetric difference notion for Boolean algebra is well-known. Matoušek introduced the orthocomplemented difference lattices (ODLs), which are ortholattices associated with a symmetric difference. He proved that the class of ODLs forms a variety. We focus on the class of all ODLs that are set-representable and prove that this class is not locally finite by constructing an infinite set-representable ODL that is generated by three elements.

In the graph theory part, we prove generating theorems and splitter theorems for 5-regular graphs. A generating theorem for a certain class of graphs tells us how to generate all graphs in this class from a few graphs by using some graph operations. A splitter theorem tells us how to build up any graph G from any graph H if G "contains" H. In this dissertation, we find generating theorems for 5-regular graphs and 5-regular loopless graphs for various edge-connectivities. We also find splitter theorems for 5-regular graphs for various edge-connectivities.

COinS