Connected matchings in special families of graphs.

A connected matching in a graph is a set of disjoint edges such that, for any pair of these edges, there is another edge of the graph incident to both of them. This dissertation investigates two problems related to finding large connected matchings in graphs. The first problem is motivated by a famous and still open conjecture made by Hadwiger stating that every k-chromatic graph contains a minor of the complete graph Kk . If true, Hadwiger\u27s conjecture would imply that every graph G has a minor of the complete graph K n/a(C), where a(G) denotes the independence number of G. For a graph G with a(G) = 2, Thomassé first noted the connection between connected matchings and large complete graph minors: there exists an ? \u3e 0 such that every graph G with a( G) = 2 contains K ?+, as a minor if and only if there exists a positive constant c such that every graph G with a( G) = 2 contains a connected matching of size cn. In Chapter 3 we prove several structural properties of a vertexminimal counterexample to these statements, extending work by Blasiak. We also prove the existence of large connected matchings in graphs with clique size close to the Ramsey bound by proving: for any positive constants band c with c \u3c ¼, there exists a positive integer N such that, if G is a graph with n =: N vertices, 0\u27( G) = 2, and clique size at most bv(n log(n) )then G contains a connected matching of size cn. The second problem concerns computational complexity of finding the size of a maximum connected matching in a graph. This problem has many applications including, when the underlying graph is chordal bipartite, applications to the bipartite margin shop problem. For general graphs, this problem is NP-complete. Cameron has shown the problem is polynomial-time solvable for chordal graphs. Inspired by this and applications to the margin shop problem, in Chapter 4 we focus on the class of chordal bipartite graphs and one of its subclasses, the convex bipartite graphs. We show that a polynomial-time algorithm to find the size of a maximum connected matching in a chordal bipartite graph reduces to finding a polynomial-time algorithm to recognize chordal bipartite graphs that have a perfect connected matching. We also prove that, in chordal bipartite graphs, a connected matching of size k is equivalent to several other statements about the graph and its biadjacency matrix, including for example, the statement that the complement of the latter contains a k x k submatrix that is permutation equivalent to strictly upper triangular matrix

The genus Gyrotoma

http://deepblue.lib.umich.edu/bitstream/2027.42/56257/1/MP012.pd

The Changing Landscape: The Urbanization of Knox County, Tennessee

Land use change, in the form of urbanization, is no longer limited to existing municipal boundaries. This change, however, is neither haphazard in direction nor unpredictable in scope. It is a series of highly evolved trends that have become one of the most highly studied and researched phenomenon in the social and physical sciences. To examine land use change as urbanization for middle tier metropolitan area, Knoxville and Knox County Tennessee were chosen as sites for my research. While Knoxville still serves as the cultural, physical and governmental center for the area, residential and commercial development have scattered throughout the county. Population and urbanization levels have, in fact, grown more rapidly outside the city limits and especially outside Knoxville\u27s Central Core Sector throughout the period covered by this research. This study, using U.S. Census data and NLCD (National Land Cover Data) information from the Environmental Protection Agency, catalogs the 2001 level of urbanization within Knox County and compares it with the 1992 level. The difference between these two was statistically assessed and a GIS based LUC (Land Use Change) model was created to project future urban growth. The projections generated by the model indicate that the majority of urban development will occur directionally within the individual sectors over time. The model also indicates that the growth within the specific sectors will fluctuate as other processes of urban change take over. This combination of factors returns a series of projections that is not only feasible but logical as well

Design and Analysis of Algorithms: Course Notes

These are my lecture notes from CMSC 651: Design and Analysis of Algorithms}, a one semester course that I taught at University of Maryland in the Spring of 1993. The course covers core material in algorithm design, and also helps students prepare for research in the field of algorithms. The reader will find an unusual emphasis on graph theoretic algorithms, and for that I am to blame. The choice of topics was mine, and is biased by my personal taste. The material for the first few weeks was taken primarily from the (now not so new) textbook on Algorithms by Cormen, Leiserson and Rivest. A few papers were also covered, that I personally feel give some very important and useful techniques that should be in the toolbox of every algorithms researcher. (Also cross-referenced as UMIACS-TR-93-72

Spartan Daily, October 18, 2005

Volume 125, Issue 30https://scholarworks.sjsu.edu/spartandaily/10173/thumbnail.jp

The Anculosae of the Alabama River drainage

http://deepblue.lib.umich.edu/bitstream/2027.42/56252/1/MP007.pd

Some results on FPGAs, file transfers, and factorizations of graphs.

by Pan Jiao Feng.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 89-93).Abstract also in Chinese.Abstract --- p.iAcknowledgments --- p.vList of Tables --- p.xList of Figures --- p.xiChapter Chapter 1. --- Introduction --- p.1Chapter 1.1 --- Graph definitions --- p.2Chapter 1.2 --- The S box graph --- p.2Chapter 1.3 --- The file transfer graph --- p.4Chapter 1.4 --- "(g, f)-factor and (g, f)-factorization" --- p.5Chapter 1.5 --- Thesis contributions --- p.6Chapter 1.6 --- Organization of the thesis --- p.7Chapter Chapter 2. --- On the Optimal Four-way Switch Box Routing Structures of FPGA Greedy Routing Architectures --- p.8Chapter 2.1 --- Introduction --- p.9Chapter 2.1.1 --- FPGA model and S box model --- p.9Chapter 2.1.2 --- FPGA routing --- p.10Chapter 2.1.3 --- Problem formulation --- p.10Chapter 2.2 --- Definitions and terminology --- p.12Chapter 2.2.1 --- General terminology --- p.12Chapter 2.2.2 --- Graph definitions --- p.15Chapter 2.2.3 --- The S box graph --- p.15Chapter 2.3 --- Properties of the S box graph and side-to-side graphs --- p.16Chapter 2.3.1 --- On the properties of the S box graph --- p.16Chapter 2.3.2 --- The properties of side-to-side graphs --- p.19Chapter 2.4 --- Conversion of the four-way FPGA routing problem --- p.23Chapter 2.4.1 --- Conversion of the S box model --- p.24Chapter 2.4.2 --- Conversion of the DAAA model --- p.26Chapter 2.4.3 --- Conversion of the DADA model --- p.27Chapter 2.4.4 --- Conversion of the DDDA model --- p.28Chapter 2.5 --- Low bounds of routing switches --- p.28Chapter 2.5.1 --- The lower bound of the DAAA model --- p.29Chapter 2.5.2 --- The lower bound of the DADA model --- p.30Chapter 2.5.3 --- The lower bound of the DDDA model --- p.31Chapter 2.6 --- Optimal structure of one-side predetermined four-way FPGA routing --- p.32Chapter 2.7 --- Optimal structures of two-side and three-side predetermined four-way FPGA routing --- p.45Chapter 2.7.1 --- Optimal structure of two-side predetermined four-way FPGA routing --- p.46Chapter 2.7.2 --- Optimal structure of three-side predetermined four-way FPGA routing --- p.47Chapter 2.8 --- Conclusion --- p.49Appendix --- p.50Chapter Chapter 3. --- "Application of (0, f)-Factorization on the Scheduling of File Transfers" --- p.53Chapter 3.1 --- Introduction --- p.53Chapter 3.1.1 --- "(0,f)-factorization" --- p.54Chapter 3.1.2 --- File transfer model and its graph --- p.54Chapter 3.1.3 --- Previous results --- p.56Chapter 3.1.4 --- Our results and outline of the chapter --- p.56Chapter 3.2 --- NP-completeness --- p.57Chapter 3.3 --- Some lemmas --- p.58Chapter 3.4 --- Bounds of file transfer graphs --- p.59Chapter 3.5 --- Comparison --- p.62Chapter 3.6 --- Conclusion --- p.68Chapter Chapter 4. --- "Decomposition Graphs into (g,f)-Factors" --- p.69Chapter 4.1 --- Introduction --- p.69Chapter 4.1.1 --- "(g,f)-factors and (g,f)-factorizations" --- p.69Chapter 4.1.2 --- Previous work --- p.70Chapter 4.1.3 --- Our results --- p.72Chapter 4.2 --- Proof of Theorem 2 --- p.73Chapter 4.3 --- Proof of Theorem 3 --- p.79Chapter 4.4 --- Proof of Theorem 4 --- p.80Chapter 4.5 --- Related previous results --- p.82Chapter 4.6 --- Conclusion --- p.84Chapter Chapter 5. --- Conclusion --- p.85Chapter 5.1 --- About graph-based approaches --- p.85Chapter 5.2 --- FPGA routing --- p.87Chapter 5.3 --- The scheduling of file transfer --- p.88Bibliography --- p.89Vita --- p.9

Extremal problems on counting combinatorial structures

The fast developing field of extremal combinatorics provides a diverse spectrum of powerful tools with many applications to economics, computer science, and optimization theory. In this thesis, we focus on counting and coloring problems in this field. The complete balanced bipartite graph on $n$ vertices has \floor{n^2/4} edges. Since all of its subgraphs are triangle-free, the number of (labeled) triangle-free graphs on $n$ vertices is at least 2^{\floor{n^2/4}}. This was shown to be the correct order of magnitude in a celebrated paper Erd\H{o}s, Kleitman, and Rothschild from 1976, where the authors furthermore proved that almost all triangle-free graphs are bipartite. In Chapters 2 and 3 we study analogous problems for triangle-free graphs that are maximal with respect to inclusion. In Chapter 2, we solve the following problem of Paul Erd\H{o}s: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. We show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among other tools the Ruzsa-Szemer\'{e}di Triangle Removal Lemma and recent results on characterizing of the structure of independent sets in hypergraphs. This is a joint work with J\'{o}zsef Balogh. In Chapter 3, we investigate the structure of maximal triangle-free graphs. We prove that almost all maximal triangle-free graphs admit a vertex partition $(X, Y)$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di Removal Lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on the characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest. This is a joint work with J\'{o}zsef Balogh, Hong Liu, and Maryam Sharifzadeh. In Chapte 4, we seek families in posets with the smallest number of comparable pairs. Given a poset $P$, a family \F\subseteq P is \emph{centered} if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the \emph{centeredness property} if for any $M$, among all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to $k$. We show that this conjecture is false for all $k\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\mathbb{F}_q^n$ has the centeredness property. Several open problems are also given. This is a joint result with J\'{o}zsef Balogh and Adam Zsolt Wagner. In Chapter 5, we consider a graph coloring problem. Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$-th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant $c$ such that for any $k$ there is a family of graphs $G$ with $\chi(G^k)$ unbounded and $\chi_{\ell}(G^k)\geq c \chi(G^k) \log \chi(G^k)$. We also provide an upper bound, $\chi_{\ell}(G^k)1$. This is a joint work with Nicholas Kosar, Benjamin Reiniger, and Elyse Yeager

Spartan Daily, November 23, 1987

Volume 89, Issue 57https://scholarworks.sjsu.edu/spartandaily/7649/thumbnail.jp