#### Abstract

Graph arrowing is concerned with determining which monochromatic subgraphs are unavoidable when coloring a given graph. There are two main avenues of research concerning arrowing: finding extremal Ramsey/Folkman graphs and categorizing the complexity of arrowing problems. Both avenues have been studied extensively for decades. In this thesis, we focus on graph arrowing problems where one of the monochromatic subgraphs being avoided is the path on three vertices, denoted as P3. Our main contributions involve computing Folkman numbers by generating graphs up to 13 vertices and proving the coNP-completeness of some arrowing problems using a novel reduction framework geared towards avoiding P3's. The (P3,H)-Arrowing Problem asks whether a given graph can be colored using two colors (red and blue) such that there are no red P3's and no blue H's, where H is a fixed graph. The few previous hardness proofs for arrowing problems relied on ad-hoc, laborious constructions of "gadgets." We introduce a general framework that can be used to prove the coNP-completeness of (P3,H)-arrowing problems. We search for gadgets computationally. These gadgets allow us to simulate variants of SAT, thus showing coNP-hardness. Finally, we use our (P3,H)-Arrowing hardness reductions to gain insight into variants of Monotone SAT. For fixed k in {4,5,6}, we show that Monotone SAT remains NP-complete under the following constraints: 1) each clause consists of exactly two unnegated literals or exactly k negated literals, 2) the variables in each clause are distinct, and 3) the number of times a variable occurs in the formula is bounded by a constant. For future work, we expect that the insight gained by our computationally assisted reductions will help us prove the complexity of other elusive arrowing problems.

#### Publication Date

4-2022

#### Document Type

Thesis

#### Student Type

Graduate

#### Degree Name

Computer Science (MS)

#### Department, Program, or Center

Computer Science (GCCIS)

#### Advisor

Edith Hemaspaandra

#### Advisor/Committee Member

Stanislaw Radziszowski

#### Advisor/Committee Member

Ivona Bezakova

#### Recommended Citation

Hassan, Zohair Raza, "Graph Arrowing: Constructions and Complexity" (2022). Thesis. Rochester Institute of Technology. Accessed from

https://scholarworks.rit.edu/theses/11142

#### Campus

RIT – Main Campus