Two-Coloring Cycles In Complete Graphs

Inspired by an investigation of Ramsey theory, this paper aims to clarify in further detail a number of results regarding the existence of monochromatic cycles in complete graphs whose edges are colored red or blue. The second half focuses on a proof given by Gyula Károlyi and Vera Rosta for the solution of all R(Cn,Ck)

Modified Ramsey Numbers

Ramsey theory is a eld of study named after the mathematician Frank P. Ramsey. In general, problems in Ramsey theory look for structure amid a collection of unstructured objects and are often solved using techniques of Graph Theory. For a typical question in Ramsey theory, we use two colors, say red and blue, to color the edges of a complete graph, and then look for either a complete subgraph of order n whose edges are all red or a complete subgraph of order m whose edges are all blue. The minimum number of vertices needed to guarantee one of these subgraphs is the Ramsey number, R(n; m). Ramsey\u27s Theorem shows that R(n; m) exists for every n and m greater than one, yet very few Ramsey numbers are known. There are many interesting modifications of the original problem such as looking for subgraphs other than complete graphs. For this thesis, we will consider modified Ramsey numbers for star graphs instead of the classical Ramsey number R(n; m). We will prove a general formula for the modified Ramsey number of two star graphs and begin exploring modified Ramsey numbers of a star graph and a path

Some Multicolor Ramsey Numbers Involving Cycles

Establishing the values of Ramsey numbers is, in general, a difficult task from the computational point of view. Over the years, researchers have developed methods to tackle this problem exhaustively in ways that require intensive computations. These methods are often backed by theoretical results that allow us to cut the search space down to a size that is within the limits of current computing capacity. This thesis focuses on developing algorithms and applying them to generate Ramsey colorings avoiding cycles. It adds to a recent trend of interest in this particular area of finite Ramsey theory. Our main contributions are the enumeration of all (C_5,C_5,C_5;n) Ramsey colorings and the study of the Ramsey numbers R(C_4,C_4,K_4) and R4(C_5)

Intrinsically linked graphs and even linking number

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where all linking numbers are even, or an n-component link with components L, A_i where lk(L,A_i) = 3k, k not 0. Links with other properties are considered as well. For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size of the complete graph is determined by the property in question.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-55.abs.htm

Graph Arrowing: Constructions and Complexity

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\u27s. 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\u27s and no blue H\u27s, 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

Modified Ramsey Numbers

Suppose you want to throw a party, but there’s a catch; you want to invite the minimum number of people to ensure there will be a group of three mutual friends or three mutual enemies, given any two people are either friends or enemies. Since you want there to be a group of three friends or three enemies, there must be at least three people invited to the party. But if you invite three people, there could easily be a situation where two people are friends while the other is an enemy. So you must invite more than three people. The same happens when looking at four or five people at the party, though; there can be a situation where there is not a group of three friends or three enemies. Now let’s consider inviting six people. If there are six people at the party, then each person will have a relationship (whether it be friends or enemies) to five other people. Let’s look at one person’s, say Lisa’s, relationships with the others at the party. If Lisa has no friends at the party, then she will be enemies with five other people. If Lisa only has one friend at the party, then she will be enemies with four other people. If she has two friends at the party, she will be enemies with three other people. Otherwise, Lisa will have three or more friends at the party. Therefore, Lisa will always either have at least three friends or at least three enemies at the party. Now let’s consider the case when Lisa has at least three friends and look at her friends’ relationships. If any two of Lisa’s friends are friends with one another, then there is a group of three friends at the party (the same goes for when she has two enemies that are enemies with one another). If none of Lisa’s three friends are friends with one another, then those friends create a group of three enemies (the same goes for when Lisa has three enemies that are all friends with one another). No matter what, there will always be a group of three mutual friends or three mutual enemies, and so we must invite at least six people to the party to ensure this occurrence

An exploration in Ramsey theory

We present several introductory results in the realm of Ramsey Theory, a subfield of Combinatorics and Graph Theory. The proofs in this thesis revolve around identifying substructure amidst chaos. After showing the existence of Ramsey numbers of two types, we exhibit how these two numbers are related. Shifting our focus to one of the Ramsey number types, we provide an argument that establishes the exact Ramsey number for h(k, 3) for k ≥ 3; this result is the highlight of this thesis. We conclude with facts that begin to establish lower bounds on these types of Ramsey numbers for graphs requiring more substructure

Ramsey Theory

The Ramsey number $R(r, b)$ is the least positive integer such that every edge 2-coloring of the complete graph $K_{R(r, b)}$ with colors red and blue either embeds a red $K_r$ or a blue $K_b$. We explore various methods to find lower bounds on $R(r,b)$, finding new results on fibrations and semicirculant graphs. Then, generalizing the Ramsey number to graphs other than complete graphs, we flesh out the missing details in the literature on a theorem that completely determines the generalized Ramsey number for cycles