A fractional approach to minimum rank and zero forcing

This thesis applies techniques from fractional graph theory to develop fractional versions of graph parameters related to minimum rank and zero forcing. Projective rank, a graph parameter with applications to quantum information, is formally related to $r$-fold generalizations of orthogonal representations for graphs. Using similar techniques, fractional minimum positive semidefinite rank is defined via $r$-fold generalizations of faithful orthogonal representations and $r$-fold minimum positive semidefinite rank, and it is shown that the fractional minimum positive semidefinite rank of any graph equals the projective rank of the complement of the graph. An alternate characterization of $r$-fold minimum positive semidefinite rank that considers the ranks of certain Hermitian matrices is also presented. Motivated by the connections between zero forcing games and minimum rank problems, an $r$-fold analogue of the positive semidefinite zero forcing process is introduced and used to define the fractional positive semidefinite forcing number of a graph. An analysis of the $r$-fold positive semidefinite forcing game leads to a three-color forcing game that allows computation of fractional positive semidefinite forcing number without appealing to the $r$-fold game. The three-color approach is applied to the standard zero forcing game and it is shown that the skew zero forcing number of a graph is exactly the parameter obtained by applying the fractionalization technique to the standard zero forcing game. Graphs whose skew zero forcing number equals zero are characterized via the three-color approach and an algorithm

Fractional Zero Forcing via Three-color Forcing Games

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.Comment: 24 page

(4, 2)-Choosability of Planar Graphs with Forbidden Structures

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

A fractional approach to minimum rank and zero forcing

This thesis applies techniques from fractional graph theory to develop fractional versions of graph parameters related to minimum rank and zero forcing. Projective rank, a graph parameter with applications to quantum information, is formally related to $r$-fold generalizations of orthogonal representations for graphs. Using similar techniques, fractional minimum positive semidefinite rank is defined via $r$-fold generalizations of faithful orthogonal representations and $r$-fold minimum positive semidefinite rank, and it is shown that the fractional minimum positive semidefinite rank of any graph equals the projective rank of the complement of the graph. An alternate characterization of $r$-fold minimum positive semidefinite rank that considers the ranks of certain Hermitian matrices is also presented. Motivated by the connections between zero forcing games and minimum rank problems, an $r$-fold analogue of the positive semidefinite zero forcing process is introduced and used to define the fractional positive semidefinite forcing number of a graph. An analysis of the $r$-fold positive semidefinite forcing game leads to a three-color forcing game that allows computation of fractional positive semidefinite forcing number without appealing to the $r$-fold game. The three-color approach is applied to the standard zero forcing game and it is shown that the skew zero forcing number of a graph is exactly the parameter obtained by applying the fractionalization technique to the standard zero forcing game. Graphs whose skew zero forcing number equals zero are characterized via the three-color approach and an algorithm.</p

Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

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\u27s problem, are discussed

Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

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.NRF (Natl Research Foundation, S’pore)Published versio