Matching Items (4)
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- All Subjects: Extremal problems (Mathematics)
- All Subjects: Graphs
- Creators: Colbourn, Charles
Description
Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
ContributorsYie, Jangwon (Author) / Czygrinow, Andrzej (Thesis advisor) / Kierstead, Henry (Committee member) / Colbourn, Charles (Committee member) / Fishel, Susanna (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2018
Description
As an example of "big data," we consider a repository of Arctic sea ice concentration data collected from satellites over the years 1979-2005. The data is represented by a graph, where vertices correspond to measurement points, and an edge is inserted between two vertices if the Pearson correlation coefficient between them exceeds a threshold. We investigate new questions about the structure of the graph related to betweenness, closeness centrality, vertex degrees, and characteristic path length. We also investigate whether an offset of weeks and years in graph generation results in a cosine similarity value that differs significantly from expected values. Finally, we relate the computational results to trends in Arctic ice.
ContributorsDougherty, Ryan Edward (Author) / Syrotiuk, Violet (Thesis director) / Colbourn, Charles (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Computer Science and Engineering Program (Contributor)
Created2015-05
Description
The primary focus of this dissertation lies in extremal combinatorics, in particular intersection theorems in finite set theory. A seminal result in the area is the theorem of Erdos, Ko and Rado which finds the upper bound on the size of an intersecting family of subsets of an n-element set and characterizes the structure of families which attain this upper bound. A major portion of this dissertation focuses on a recent generalization of the Erdos--Ko--Rado theorem which considers intersecting families of independent sets in graphs. An intersection theorem is proved for a large class of graphs, namely chordal graphs which satisfy an additional condition and similar problems are considered for trees, bipartite graphs and other special classes. A similar extension is also formulated for cross-intersecting families and results are proved for chordal graphs and cycles. A well-known generalization of the EKR theorem for k-wise intersecting families due to Frankl is also considered. A stability version of Frankl's theorem is proved, which provides additional structural information about k-wise intersecting families which have size close to the maximum upper bound. A graph-theoretic generalization of Frankl's theorem is also formulated and proved for perfect matching graphs. Finally, a long-standing conjecture of Chvatal regarding structure of maximum intersecting families in hereditary systems is considered. An intersection theorem is proved for hereditary families which have rank 3 using a powerful tool of Erdos and Rado which is called the Sunflower Lemma.
ContributorsKamat, Vikram M (Author) / Hurlbert, Glenn (Thesis advisor) / Colbourn, Charles (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Kierstead, Henry (Committee member) / Arizona State University (Publisher)
Created2011
Description
Polar ice masses can be valuable indicators of trends in global climate. In an effort to better understand the dynamics of Arctic ice, this project analyzes sea ice concentration anomaly data collected over gridded regions (cells) and builds graphs based upon high correlations between cells. These graphs offer the opportunity to use metrics such as clustering coefficients and connected components to isolate representative trends in ice masses. Based upon this analysis, the structure of sea ice graphs differs at a statistically significant level from random graphs, and several regions show erratically decreasing trends in sea ice concentration.
ContributorsWallace-Patterson, Chloe Rae (Author) / Syrotiuk, Violet (Thesis director) / Colbourn, Charles (Committee member) / Montgomery, Douglas (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Computer Science and Engineering Program (Contributor)
Created2013-05