Thesis (Ph.D.)--University of Central Florida, 2009. / Adviser: Yue Zhao. Includes bibliographical references (p. 102-104).

Thesis (M. S.)--Computing, Georgia Institute of Technology, 2009. / Committee Chair: Prof. Richard Lipton; Committee Member: Prof. Dana Randall; Committee Member: Prof. H. Venkateswaran. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Thesis (M.S.)--School of Mathematics, Georgia Institute of Technology, 2005. Directed by Xingxing Yu. / Xingxing Yu, Committee Chair ; Robin Thomas, Committee Member ; Prasad Tetali, Committee Member ; Anurag Singh, Committee Member. Includes bibliographical references.

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Hajós conjectured that, for any positive integer k, every graph containing no K_(k+1)-subdivision is k-colorable. This is true when k is at most three, and false when k exceeds six. Hajós' conjecture remains open for k=4,5. We will first present some known results on Hajós' conjecture. Then we derive a result on the structure of 2-connected graphs with no cycle through three specified vertices. This result will then be used for the proof of the main result of this thesis. We show that any possible counterexample to Hajós' conjecture for k=4 with minimum number of vertices must be 4-connected. This is a step in an attempt to reduce Hajós' conjecture for k=4 to the conjecture of Seymour that any 5-connected non-planar graph contains a K_5-subdivision.

Thesis (M.S.)--Ohio University, March, 2004. / Title from PDF t.p. Includes bibliographical references (leaves 51-55).

Thesis--Auburn University, 2009. / Abstract. Vita. Includes bibliographical references (p. 21).

abstract: The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum number of colors needed to color $V(G)$ such that no adjacent vertices receive the same color. The coloring number $\col(G)$ of a graph $G$ is the minimum number $k$ such that there exists a linear ordering of $V(G)$ for which each vertex has at most $k-1$ backward neighbors. It is well known that the coloring number is an upper bound for the chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is a generalization of the coloring number, and it was first introduced by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$ is the minimum integer $k$ such that for some linear ordering $L$ of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller vertices $u$ (with respect to $L$) with a path of length at most $r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$. The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$ is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$ if and only if the distance between $x$ and $y$ in $G$ is $3$. This dissertation improves the best known upper bound of the chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$ of planar graphs $G$, which is $105$, to $95$. It also improves the best known lower bound, which is $7$, to $9$. A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2020

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