## Electronic Theses and Dissertations

#### Title

Problems in Extremal and Ramsey Graph Theory

1364

2015

4-21-2015

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematical Sciences

Mathematics

Bela Bollobas

Paul Balister

Anna Kaminska

#### Abstract

Chapter 1 is dedicated to the results in the papers with Bhargav P. Narayanan. Given an edge coloring of the complete graph on N, we say that a subset of N is exactly m-colored if exactly m colors appear inside the subset. We answer many of the questions about finding exactly m-colored subgraphs. In Chapter 2, we present joint work with Bela Bollobas, Bhargav P. Narayanan and Alexander D. Scott. Let us say that a graph is splittable if the vertices can be partitioned into two equal halves such that each half induces the same number of edges. The main result is that any graph of order n can be made splittable by deleting at most o(n) vertices. This answers a question of Caro and Yuster. Chapter 3 is based on joint work with Victor Falgas-Ravry, Daniel Korandi, Shoham Letzter and Bhargav P. Narayanan. A set is separated by a collection of its subsets if any two elements of the set can be distinguished using some subset in the collection. We consider a question of separating the edge set of a graph using only paths. We conjecture that every graph of order n admits a separating path system of size linear in n and prove this in certain interesting special cases including random graphs and graphs with linear minimum degree. Chapter 4 presents results from a joint paper with Gabor Meszaros. A triple of vertices in a graph is a frustrated triangle if it induces an odd number of edges. We study the set F_n of possible number of frustrated triangles f(G) in a graph G on n vertices. Our main result is that F_n contains two interlacing sequences 0=a_0<=b_0<=a_1<=b_1<=...<=a_m<=b_m~n^{3/2} such that the intersection of F_n and (b_t,a_{t+1}) is empty for all t, where the gaps are |b_t-a_{t+1}|=(n-2)-t(t+1) and |a_t-b_t|=t(t-1).