Year of Award
Doctor of Philosophy (PhD)
Department or School/College
Department of Mathematical Sciences
P. Mark Kayll, Cory Palmer
Eric Chesebro, Nikolaus Vonessen, Richard Bridges
Combinatorics, Graph Theory, Random Graphs
University of Montana
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.
Finlay, John Arthur, "Randomly Perturbed Graphs and Rainbow Connectivity" (2022). Graduate Student Theses, Dissertations, & Professional Papers. 11935.
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