Georgia Institute of Technology College of Sciences

This work studies two topics in sequence analysis. In the first part, we investigate the large deviations of the shape of the random RSK Young diagrams, associated with a random word of size n whose letters are independently drawn from an alphabet of size m=m(n). When the letters are drawn uniformly and when both n and m converge together to infinity, m not growing too fast with respect to n, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. Since the length of the top row of the Young diagrams is the length of the longest (weakly) increasing subsequence of the random word, the corresponding large deviations follow. When the letters are drawn with non-uniform probability, a control of both highest probabilities will ensure that the length of the top row of the diagrams satisfies a large deviation principle. In either case, speeds and rate functions are identified. To complete this first part, non-asymptotic concentration bounds for the length of the top row of the diagrams are obtained. In the second part, we investigate the order of the r-th, 1\le r < +\infty, central moment of the length of the longest common subsequence of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, the r-th central moment is shown to be of order n^{r/2}. In particular, when r=2, the order of the variance is linear.

Details can be found at http://www.aco.gatech.edu/dissert/postle.html.

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.