(Georgia Institute of Technology, 2008-08-25)
Bilinski, Mark
Jackson and Wormald show that every 3-connected
K_1,d-free graph, on n vertices, contains a cycle of length at least 1/2 n^g(d) where g(d) = (log_2 6 + 2 log_2 (2d+1))^-1. For d = 3, g(d) ~ 0.122.
Improving this bound, we prove that if G is a 3-connected claw-free graph on at least 6 vertices, then there exists a cycle C in G such that |E(C)| is at least c n^g+5, where
g = log_3 2 and c > 1/7 is a constant.
To do this, we instead prove a stronger theorem that requires the cycle to contain two specified edges. We then use Tutte decomposition to partition the graph and then use the inductive
hypothesis of our theorem to find paths or cycles in the different parts of the decomposition.
(Georgia Institute of Technology, 2006-11-20)
Jiang, Wen
We study codes that have identifiable parent property. Such codes are called IPP codes. Research on IPP codes is motivated by design of
schemes that protect against piracy of digital products.
Construction and decoding of maximum IPP codes have been studied in rich literature. General bounds on F(n,q), the maximum size of IPP
codes of length n over an alphabet with q elements, have been obtained through the use of techniques from graph theory and combinatorial
design. Improved bounds on F(3,q) and F(4,q) are obtained. Probabilistic techniques are also used to prove the existence of certain IPP codes.
We prove a precise formula for F(3,q), construct maximum IPP codes with size F(3,q), and give an efficient decoding algorithm for such codes. The main techniques used in this thesis are from graph theory and nonlinear optimization. Our approach may be used to improve bounds on F(2k+1, q). For
example, we characterize the associated graphs of
maximum IPP codes of length 5, and obtain bounds on F(5,q).
(Georgia Institute of Technology, 2004-05-20)
Zickfeld, Florian
Hajós conjectured that, for any positive integer
k, every graph containing no K_(k+1)-subdivision is k-colorable. This is true when k is at most three, and false when k exceeds six. Hajós' conjecture remains open for k=4,5.
We will first present some known results on Hajós' conjecture. Then we derive a result on the structure of 2-connected graphs
with no cycle through three specified vertices. This result will then be used for the proof of the main result of this thesis. We show that any possible counterexample to Hajós' conjecture for
k=4 with minimum number of vertices must be 4-connected. This is
a step in an attempt to reduce Hajós' conjecture for k=4 to the conjecture of Seymour that any 5-connected non-planar graph contains a K_5-subdivision.