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The study of chromaticity of complete tripartite graphs
Author
Ng, Boon Leong
Supervisor
Dong, F. M.
Abstract
In this thesis, I first present a brief history of research in chromatic polynomials, starting from their initial formulation by Birkhoff [5] to current research, especially in the areas of chromatic uniqueness and chromatic equivalence [38, 39]. The thesis concludes with an exploration of a particular family of graphs, the complete tripartite graphs, and with the main result of this thesis, the determination of the chromatic equivalence class of the graph K1,n,n+k, where 0 ≥ k ≥ 2 in Theorems 3.10, 3.11 and 3.13.
Chapter 1 takes the reader through a historical review of the development of the theory of chromatic polynomials, some methods of calculating chromatic polynomials, and some fundamental results relating the degree, coe cients, and roots of the chromatic polynomial to various graph parameters.
Chapter 2 introduces the related notions of chromatic equivalence and chromatic uniqueness, as well as some results regarding the chromatic equivalence and uniqueness of three families of graphs - wheels, θ-graphs, and complete bipartite and multipartite graphs - along with some other graphs which are closely related to them. Some recent work on Tutte polynomial equivalence is also mentioned.
Chapter 3 investigates complete tripartite graphs in detail, reviewing the recent research into showing that various classes of complete tripartite graphs are chromatically unique, and the recent work by Gek Ling Chia and Chee Kit Ho [18, 19] in determining the chromatic equivalence classes of K1,n,n and K1,n,n+1. Building on their work, the main result of this thesis, Theorem 3.13, in which the chromatic equivalence class of K1,n,n+2 is determined, is presented, and some possible avenues of future research are also discussed.
Chapter 1 takes the reader through a historical review of the development of the theory of chromatic polynomials, some methods of calculating chromatic polynomials, and some fundamental results relating the degree, coe cients, and roots of the chromatic polynomial to various graph parameters.
Chapter 2 introduces the related notions of chromatic equivalence and chromatic uniqueness, as well as some results regarding the chromatic equivalence and uniqueness of three families of graphs - wheels, θ-graphs, and complete bipartite and multipartite graphs - along with some other graphs which are closely related to them. Some recent work on Tutte polynomial equivalence is also mentioned.
Chapter 3 investigates complete tripartite graphs in detail, reviewing the recent research into showing that various classes of complete tripartite graphs are chromatically unique, and the recent work by Gek Ling Chia and Chee Kit Ho [18, 19] in determining the chromatic equivalence classes of K1,n,n and K1,n,n+1. Building on their work, the main result of this thesis, Theorem 3.13, in which the chromatic equivalence class of K1,n,n+2 is determined, is presented, and some possible avenues of future research are also discussed.
Date Issued
2014
Call Number
QA166 Ng
Date Submitted
2014