A study of saturation number

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Author
Burr, Erika
Chair
Faudree, Jill
Committee
Williams, Gordon
Berman-Williams, Leah
Keyword
Extremal problems (Mathematics)
Graph theory
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URI
http://hdl.handle.net/11122/7864
Abstract
This paper seeks to provide complete proofs in modern notation of (early) key saturation number results and give some new results concerning the semi-saturation number. We highlight relevant results from extremal theory and present the saturation number for the complete graph Kk; and the star K₁,t, elaborating on the proofs provided in the 1964 paper A Problem in Graph Theory by Erdos, Hajnal and Moon and the 1986 paper Saturated Graphs with Minimal Number of Edges by Kászonyi and Tuza. We discuss the proof of a general bound on the saturation number for a family of target graphs provided by Kászonyi and Tuza. A discussion of related results showing that the complete graph has the maximum saturation number among target graphs of the same order and that the star has the maximum saturation number among target trees of the same order is included. Before presenting our result concerning the semi-saturation number for the path Pk; we discuss the structure of some Pk-saturated trees of large order as well as the saturation number of Pk with respect to host graphs of large order.
Description
Thesis (M.S.) University of Alaska Fairbanks, 2017
Table of Contents
Chapter 1: Introduction -- 1.1 Basic definitions -- 1.2 Saturation number -- 1.3 Chapter overview. Chapter 2: A brief history of saturation number -- 2.1 Extremal theory -- 2.2 The minimal Kk-saturated Graph: Ak(n). Chapter 3: General saturation number results -- 3.1 General bounds for sat(n,F) -- 3.2 Stars. Chapter 4: Saturation numbers for paths and other families of trees -- 4.1 Isolated edges -- 4.2 Paths -- 4.3 Trees of minimum saturation number -- 4.4 Other tree saturation number results -- 4.4.1 Properties of subtrees and saturation number bounds -- 4.4.2 Trees T for which there exists a minimal T-saturated forest. Chapter 5: Semi-saturation number -- 5.1 Motivation -- 5.2 The semi-saturation number for Pk. Chapter 6: Further questions -- References.
Date
2017-08
Type
Thesis
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