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Item6-connected graphs are two-three linked(Georgia Institute of Technology, 2019-11-11) Xie, ShijieLet $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on Four Color Theorem, Hadwiger's conjecture for $K_6$, and J\o rgensen's conjecture, Robertson and Seymour asked when does $G$ contain disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We prove that if $G$ is 6-connected then such $G_1,G_2$ exist. Joint work with Robin Thomas and Xingxing Yu.
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ItemColoring graphs with no k5-subdivision: disjoint paths in graphs(Georgia Institute of Technology, 2019-03-27) Xie, QiqinThe Four Color Theorem states that every planar graph is 4-colorable. Hajos conjectured that for any positive integer k, every graph containing no K_{k+1}-subdivision is k-colorable. However, Catlin disproved Hajos conjecture for k>=6. It is not hard to prove that the conjecture is true for k<=3. Hajos' conjecture remains open for k=4 and k=5. We consider a minimal counterexample to Hajos conjecture for k=4. We use Hajos graph to denote such counterexample. One important step to understand graphs containing K5-subdivisions is to solve the topological H problem. We characterize graphs with no topological H, and the characterization is used by He, Wang, and Yu to show that graph containing no K5-subdivisions is planar or has a 4-cut, establishing conjecture of Kelmans and Seymour. Besides the topological H problem, we also obtained some further structural information of Hajos graphs.
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ItemSubdivisions of complete graphs(Georgia Institute of Technology, 2017-05-23) Wang, YanA subdivision of a graph G, also known as a topological G and denoted by TG, is a graph obtained from G by replacing certain edges of G with internally vertex-disjoint paths. This dissertation studies a problem in structural graph theory regarding subdivisions of a complete graph in graphs. In this dissertation, we focus on TK_5, or subdivisions of K_5. A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K_5 or K_{3,3}. Wagner proved in 1937 that if a graph other than K_5 does not contain any subdivision of K_{3,3} then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K_5 then it is planar or it admits a cut of size at most 4. In this dissertation, we give a proof of the Kelmans-Seymour conjecture.
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ItemStatistical inference for some risk measures(Georgia Institute of Technology, 2017-05-03) Hou, YanxiRecently, many risk measures have been developed for various types of risk based on multiple financial variables. However, statistical properties of these risk measures are not fully understood, and there are very few effective inference methods for them in applications to financial data. This thesis addresses asymptotic behaviors and statistical inference methods for several newly proposed risk measures, including relative risk and conditional value-at-risk. These risk metrics are intended to measure the tail risks and/or systemic risk in financial markets. We consider conditional Value-at-Risk based on a linear regression model. We extend the assumptions on predictors and errors of the model, which make the model more flexible for the financial data. We then consider a relative risk measure based on a benchmark variable. The relative risk measure is proposed as a monitoring index for systemic risk of financial system. We also propose a new tail dependence measure based on the limit of conditional Kendall’s tau. The new tail dependence can be used to distinguish between the asymptotic independence and dependence in extreme value theory. For asymptotic results of these measures, we derive both normal and Chi-squared approximations. These approximations are a basis for inference methods. For normal approximation, the asymptotic variances are too complicated to estimate due to the complex forms of risk measures. Quantifying uncertainty is a practical and important issue in risk management. We propose several empirical likelihood methods to construct interval estimation based on Chi-squared approximation. Simulation study and real data analysis illustrate the usefulness of these risk measures and our inference methods. In particular, the empirical likelihood methods are very effective and easy to implement for practical applications.
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ItemSpecial TK5 in graphs containing K4-(Georgia Institute of Technology, 2017-04-05) He, DaweiGiven a graph K, TK is used to denote a subdivision of K, which is a graph obtained from K by substituting some edges for paths. The well-known Kelmans-Seymour conjecture states that every nonplanar 5-connected graph contains TK5 . Ma and Yu proved the conjecture for graphs containing K4-. In this dissertation, we strengthen their result in two ways. The results will be useful for completely resolving the Kelmans-Seymour conjecture. Let G be a 5-connected nonplanar graph and let x1, x2, y1, y2 in V(G) be distinct, such that G[{x1, x2, y1, y2}] is isomorphic to K4- and y1y2 is not in E(G). We show that one of the following holds: G - y2 contains K4-, or G contains a TK5 in which y2 is not a branch vertex, or G has a special 5-separation, or for any distinct w1, w2, w3 in N(y2) - {x1, x2}, G - {y2v : v not in {x1, x2, w1, w2, w3}} contains TK5. We show that one of the following holds: G - x1 contains K4-, or G contains a TK5 in which x1 is not a branch vertex, or G contains a K4- in which x1 is of degree 2, or {x2, y1, y2} may be chosen so that for any distinct z0, z1 in N(x1) - {x2, y1, y2}, G - {x1v : v not in {z0, z1, x2, y1, y2}} contains TK5.
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ItemForbidden subgraphs and 3-colorability(Georgia Institute of Technology, 2012-06-26) Ye, TianjunClassical vertex coloring problems ask for the minimum number of colors needed to color the vertices of a graph, such that adjacent vertices use different colors. Vertex coloring does have quite a few practical applications in communication theory, industry engineering and computer science. Such examples can be found in the book of Hansen and Marcotte. Deciding whether a graph is 3-colorable or not is a well-known NP-complete problem, even for triangle-free graphs. Intuitively, large girth may help reduce the chromatic number. However, in 1959, Erdos used the probabilitic method to prove that for any two positive integers g and k, there exist graphs of girth at least g and chromatic number at least k. Thus, restricting girth alone does not help bound the chromatic number. However, if we forbid certain tree structure in addition to girth restriction, then it is possible to bound the chromatic number. Randerath determined several such tree structures, and conjectured that if a graph is fork-free and triangle-free, then it is 3-colorable, where a fork is a star K1,4 with two branches subdivided once. The main result of this thesis is that Randerath’s conjecture is true for graphs with odd girth at least 7. We also give a proof that Randerath’s conjecture holds for graphs with maximum degree 4.
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ItemJudicious partitions of graphs and hypergraphs(Georgia Institute of Technology, 2011-05-04) Ma, JieClassical partitioning problems, like the Max-Cut problem, ask for partitions that optimize one quantity, which are important to such fields as VLSI design, combinatorial optimization, and computer science. Judicious partitioning problems on graphs or hypergraphs ask for partitions that optimize several quantities simultaneously. In this dissertation, we work on judicious partitions of graphs and hypergraphs, and solve or asymptotically solve several open problems of Bollobas and Scott on judicious partitions, using the probabilistic method and extremal techniques.