Abstract. Fractional minimum positive semidefinite rank is defined from r-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An r-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces r-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theory, including Tsirelson's problem, are discussed.

An r-fold analogue of the positive semidefinite zero forcing process that is carried out on the r-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any ℓ ∈ {3, 4, 5, 6, 7}, a planar graph is 4-choosable if it is ℓ-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ ∈ {5, 6, 7} that a planar graph is (4, 2)-choosable if it does not contain chorded ℓ-cycles.

I did it! Thanks for all of the love and support that you've given me over the years, especially while I've been away at grad school. I've learned a lot at Iowa State, but perhaps the most important lesson of all is that our family is amazing and we're all darned lucky to have each other. No one deserves this dedication as much as you guys.