## Electronic Theses and Dissertations

2019

1-1-2019

Dissertation

#### Degree Name

Doctor of Philosophy

#### Department

Mathematical Sciences

#### Committee Member

David Grynkiewicz

Anna Kaminska

#### Abstract

This dissertation answers problems in three areas of combinatorics - processes on graphs, graph coloring, and antichains in a partially ordered set.First we consider Zero Forcing on graphs, an iterative infection process introduced by AIM Minimum Rank - Special Graphs Workgroup in 2008. The Zero Forcing process is a graph infection process obeying the following rules: a white vertex is turned black if it is the only white neighbor of some black vertex. The Zero Forcing Number of a graph is the minimum cardinality over all sets of black vertices such that, after a finite number of iterations, every vertex is black. We establish some results about the zero forcing number of certain graphs and provide a counter example of a conjecture of Gentner and Rautenbach. This chapter is joint with Gabor Meszaros, Antonio Girao, and Chapter 3 appears in Discrete Math, Vol. 341(4).In the second part, we consider problems in the area of Dynamic Coloring of graphs. Originally introduced by Montgomery in 2001, the r-dynamic chromatic number of a graph G is the least k such that V(G) is properly colored, and each vertex is adjacent to at least r different colors. In this coloring regime, we prove some bounds for graphs with lattice like structures, hypercubes, generalized intervals, and other graphs of interest. Next, we establish some of the first results in the area of r-dynamic coloring on random graphs. The work in this section is joint with Peter van Hintum.In the third part, we consider a question about the structure of the partially ordered set of all connected graphs. Let G be the set of all connected graphs on vertex set [n]. Define the partial ordering < on G as follows: for G,H G let G < H if E(G) E(H). The poset (G,