Constrained Ramsey Numbers

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

A study of the total coloring of graphs.

The area of total coloring is a more recent and less studied area than vertex and edge coloring, but recently, some attention has been given to the Total Coloring Conjecture, which states that each graph\u27s total chromatic number xT is no greater than its maximum degree plus two. In this dissertation, it is proved that the conjecture is satisfied by those planar graphs in which no vertex of degree 5 or 6 1ies on more than three 3-cycles. The total independence number aT is found for some families of graphs, and a relationship between that parameter and the size of a graph\u27s minimum maximal matching is discussed. For colorings with natural numbers, the total chromatic sum ST is introduced, as is total strength (oT of a graph. Tools are developed for proving that a total coloring has minimum sum, and this sum is found for some graphs including paths, cycles, complete graphs, complete bipartite graphs, full binary trees, and some hypercubes. A family of graphs is found for which no optimal total coloring maximizes the smallest color class. Lastly, the relationship between a graph\u27s total chromatic number and its total strength is explored, and some graphs are found that require more than their total chromatic number of colors to obtain a minimum sum

Linear-Time Algorithms for Edge-Based Problems

There is a dearth of algorithms that deal with edge-based problems in trees, specifically algorithms for edge sets that satisfy a particular parameter. The goal of this thesis is to create a methodology for designing algorithms for these edge-based problems. We will present a variant of the Wimer method [Wimer et al. 1985] [Wimer 1987] that can handle edge properties. We call this variant the Wimer edge variant. The thesis is divided into three sections, the first being a chapter devoted to defining and discussing the Wimer edge variant in depth, showing how to develop an algorithm using this variant, and an example of this process, including a run of an algorithm developed using this method. The second section involves algorithms developed using the Wimer edge variant. We will provide algorithms for a variety of edge parameters, including four different matching parameters (connected, disconnected, induced and 2-matching), three different domination parameters (edge, total edge and edge-vertex) and two covering parameters (edge cover and edge cover irredundance). Each of these algorithms are discussed in detail and run in linear time. The third section involves an attempt to characterize the Wimer edge variant. We show how the variant can be applied to three classes of graphs: weighted trees, unicyclic graphs and generalized series-parallel graphs. For each of these classes, we detail what adaptations are required (if any) and design an algorithm, including showing a run on an example graph. The fourth chapter is devoted to a discussion of what qualities a parameter has to have in order to be likely to have a solution using the Wimer edge variant. Also in this chapter we discuss classes of graphs that can utilize the Wimer edge variant. Other topics discussed in this thesis include a literature review, and a discussion of future work. There are plenty of options for future work on this topic, which hopefully this thesis can inspire. The intent of this thesis is to provide the foundation for future algorithms and other work in this area

Algorithms for Fundamental Problems in Computer Networks.

Traditional studies of algorithms consider the sequential setting, where the whole input data is fed into a single device that computes the solution. Today, the network, such as the Internet, contains of a vast amount of information. The overhead of aggregating all the information into a single device is too expensive, so a distributed approach to solve the problem is often preferable. In this thesis, we aim to develop efficient algorithms for the following fundamental graph problems that arise in networks, in both sequential and distributed settings. Graph coloring is a basic symmetry breaking problem in distributed computing. Each node is to be assigned a color such that adjacent nodes are assigned different colors. Both the efficiency and the quality of coloring are important measures of an algorithm. One of our main contributions is providing tools for obtaining colorings of good quality whose existence are non-trivial. We also consider other optimization problems in the distributed setting. For example, we investigate efficient methods for identifying the connectivity as well as the bottleneck edges in a distributed network. Our approximation algorithm is almost-tight in the sense that the running time matches the known lower bound up to a poly-logarithmic factor. For another example, we model how the task allocation can be done in ant colonies, when the ants may have different capabilities in doing different tasks. The matching problems are one of the classic combinatorial optimization problems. We study the weighted matching problems in the sequential setting. We give a new scaling algorithm for finding the maximum weight perfect matching in general graphs, which improves the long-standing Gabow-Tarjan's algorithm (1991) and matches the running time of the best weighted bipartite perfect matching algorithm (Gabow and Tarjan, 1989). Furthermore, for the maximum weight matching problem in bipartite graphs, we give a faster scaling algorithm whose running time is faster than Gabow and Tarjan's weighted bipartite {it perfect} matching algorithm.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113540/1/hsinhao_1.pd