## Presentation Type

Oral Presentation

## Category

STEM (science, technology, engineering, mathematics)

## Abstract/Artist Statement

In Graph Theory we describe an object called a graph G(V,E) which is a set of vertices, V, and the set of edges, E, between the vertices. Practically any type of system which can be described by a network, such as the interaction between neurons in the brain, the connections within a crime ring, or species migration between islands in the South Pacific, can be modeled at some level by a graph. Within theoretical mathematics they provide an enormous field of study in combinatorics with applications in geometry, number theory, and probability.

One graph model we explore in our research is that of a randomly perturbed graph. This model begins with an arbitrarily dense graph with minimum degree d, where degree is the number of edges incident to each vertex. To this graph we add m additional edges to get our final graph G. We then randomly color edges with r colors. Based on a conjecture by Anastos and Frieze from 2019, we have shown that with r=5 and m constant, G is rainbow connected with high probability. A graph is connected if there is an edge path between any two vertices. A graph is rainbow connected if there exists a path between any two vertices where no color is repeated along the path.

We relied heavily on the probabilistic method to prove our statement. The probabilistic method lies at the junction of discrete math and probability theory. The method involves proving the existence of a structure with desired properties by defining an appropriate sample space of structures and showing that the a structure with desired properties exists in the sample space with positive probability. Thinking of it conversely, the probability that the structure with the desired properties does not exist is less than one in violation of Kolmagrov's axioms.

This presentation will introduce the notions of Graph Theory required to understand our result. Additionally, it will be an introduction to the probabilistic method which is likely to be novel to people who are not specialized in the area, but has increasingly shown its far reaching applications. Finally, this presentation will show the use of the probabilistic method within our argument. My hope is that this will provide a general audience an understanding of our result and the method used to prove it without straying too far into the thicket of discipline-specific details.

## Mentor Name

Cory Palmer

A Solution To An Open Problem In Random Graph Theory

UC 331

In Graph Theory we describe an object called a graph G(V,E) which is a set of vertices, V, and the set of edges, E, between the vertices. Practically any type of system which can be described by a network, such as the interaction between neurons in the brain, the connections within a crime ring, or species migration between islands in the South Pacific, can be modeled at some level by a graph. Within theoretical mathematics they provide an enormous field of study in combinatorics with applications in geometry, number theory, and probability.

One graph model we explore in our research is that of a randomly perturbed graph. This model begins with an arbitrarily dense graph with minimum degree d, where degree is the number of edges incident to each vertex. To this graph we add m additional edges to get our final graph G. We then randomly color edges with r colors. Based on a conjecture by Anastos and Frieze from 2019, we have shown that with r=5 and m constant, G is rainbow connected with high probability. A graph is connected if there is an edge path between any two vertices. A graph is rainbow connected if there exists a path between any two vertices where no color is repeated along the path.

We relied heavily on the probabilistic method to prove our statement. The probabilistic method lies at the junction of discrete math and probability theory. The method involves proving the existence of a structure with desired properties by defining an appropriate sample space of structures and showing that the a structure with desired properties exists in the sample space with positive probability. Thinking of it conversely, the probability that the structure with the desired properties does not exist is less than one in violation of Kolmagrov's axioms.

This presentation will introduce the notions of Graph Theory required to understand our result. Additionally, it will be an introduction to the probabilistic method which is likely to be novel to people who are not specialized in the area, but has increasingly shown its far reaching applications. Finally, this presentation will show the use of the probabilistic method within our argument. My hope is that this will provide a general audience an understanding of our result and the method used to prove it without straying too far into the thicket of discipline-specific details.