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

Publisher

University of Montana

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.

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© Copyright 2022 John Arthur Finlay