Georgia Institute of Technology College of Sciences

Classical 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. Intutively, 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 an outline of a proof that Randerath's conjecture holds for graphs with maximum degree 4.

We discuss the theory of symmetric Groebner bases, a concept allowing one to prove Noetherianity results for symmetric ideals in polynomial rings with an infinite number of variables. We also explain applications of these objects to other fields such as algebraic statistics, and we discuss some methods for computing with them on a computer. Some of this is joint work with Matthias Aschenbrener and Seth Sullivant.

We derive optimal strategies for a pursuit-and-evasion game and show that when pitted against each other, the two strategies construct a small set containing unit-length line segments at all angles. Joint work with Y. Babichenko, Y. Peres, R. Peretz, and P. Sousi.

Over the past 30 years, researchers have developed successively faster algorithms for the maximum flow problem. The best strongly polynomial time algorithms have come very close to O(nm) time. Many researchers have conjectured that O(nm) time is the "true" worst case running time. We resolve the issue in two ways. First, we show how to solve the max flow problem in O(nm) time. Second, we show that the running time is even faster if m = O(n). In this case, the running time is O(n^2/log n).