Year of Award

2022

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Co-chair

P. Mark Kayll, Cory Palmer

Commitee Members

Eric Chesebro, Nikolaus Vonessen, Richard Bridges

Keywords

Combinatorics, Graph Theory, Random Graphs

Abstract

In this work we explore randomly perturbed graphs; that is, for an arbitrarily dense graph H we add a set R consisting of m edges randomly to create a graph G. We then randomly color the edges of G with r colors. We prove, for r ≥ 5 and m a large enough constant, that between any two vertices in G there exists a rainbow path and thus G is rainbow connected. This result confirms a conjecture of Anastos and Frieze [How many randomly colored edges make a randomly colored dense graph rainbow Hamiltonian or rainbow connected?, J. Graph Theory 92 (2019), no. 4, 405–414] which resolved the case when r ≥ 7 and m is a function of n (that tends to infinity arbitrarily slowly). We also explore concepts and results related to this result.

Download

Share

COinS
 

© Copyright 2022 John Arthur Finlay