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\textbf {\large ORAL EXAM}\\ 
$ $\\
December 16, 2003\\
Mohamud Mohammed\\ 
email: mohamudm@math.rutgers.edu \\ 
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Committee: J. Beck, J. Koml\'os, M. Saks and D. Zeilberger \\
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\noindent \textbf {\large Experimental Mathematics, Combinatorics and Graph theory}\\
\begin{enumerate}
	\item \textbf {Experimental Mathematics and Combinatorics}
	\begin {description}
	\item[Hypergeometric Functions:] definition and representation, notion of q-analog and examples.%\\
	\item[The five basic algorithms for hypergeometric identities:] Mary Celine Fasenmyer's algorithm, Gosper's algorithm, Zeilberger's algorithm, Petkovs\'ek's algorithm, the WZ phenomena, Dual(Shadow) and the Companion identity and the Fundamental Theorem for hypergeometric identities.%\\
	\item[The (Umbral) Transfer-Matrix-Method and applications to enumeration:] Finite and infinite case.%\\
	\item[Integer relation algorithms and applications:] The LLL based Algorithm.%\\
	\item[Basic enumeration:] Counting arguments, Generating functions, The Lagrange inversion formula, P-recursive sequences, Stirling numbers, Bell numbers, Catalan numbers and Inclusion-Exclusion Principle.%\\     
	\end {description}
\item \textbf {Graph Theory }
	\begin {description}
	\item[The basics of Graph theory:] graph parameters, trees, bipartite graphs, Eulerian tour, paths and cycles.%\\
	\item[Matching:] K\"onig's theorem, Hall's theorem, Tutte's theorem, The Gallai and Milgram theorem, Path covers and Dilworth's theorem.%\\
	\item[Connectivity:] the structure of 2-connected and 3-connected graphs, k-connected graphs, Menger's theorem, max-flow-min-cut theorem, statement of Mader's theorem, Edge-disjoint spanning trees and paths between given pair of points.%\\
	\item[Planar graphs:] Euler's formula, Kuratowski's theorem, plane duality, and abstract duality.%\\
	\item[Coloring:] Vertex coloring, Edge coloring, Five color theorem, Brook's theorem, K\"onig's theorem, Vizing's theorem, Weak perfect graph theorem, Strong perfect graph theorem and Lov\v asz's theorem.%\\
	\item[Probabilistic Method (Random graphs):] Stirling's formula, G(n,p) verses G(n,M), monotone properties, properties of almost all graphs, Erd\"os and Reyni's theorem, threshold function for having a certain graph as a subgraph,  relationship between being connected and having no isolated vertices.%\\
	\item[Hamiltonian Cycles:] Dirac's theorem, Ore's theorem, Hamiltonian and degree sequences(Chv\'atal's theorem).%\\
	\item[Extremal problems:] Tur\'an's theorem, statement of  regularity lemma and how it is  applied.%\\
	\item[Minors, Trees, and WQO:] Well-quasi ordering, the graph minor theorem for trees, the graph minor theorem and implications for embedding graphs on surfaces.%\\
	\item[Matroids:] Definitions of matroids, The Matroid Intersection theorem, The Matroid Sum theorem, The packing theorem, The covering theorem and applications to graph theory.   
	\end {description}
\item \textbf {Probability theory}\\
	Probability space, random-variable, linearity of expectation, conditional probability, Chebyshev's inequality, Markov's inequality, the Lov\'asz local lemma, Central limit theorem, Law of large numbers.%\\
	
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\item \textbf {Ramsey theory}\\ 
	Definition and existence of Ramsey numbers, Original Ramsey Theorem, Infinite Ramsey theorem, Ramsey numbers: $R(3,3)$,$R(3,4)$,$R(4,4)$,$R(2,k)$ Chv\'atal-R\"odl-%\\
Szemer\'edi-Trotter theorem, Bounds on the Ramsey numbers, ven de Waerden %\\
theorem, Schurs triples, R\"oth's theorem for arithematic progression(statement) and the statement of  Szemer\'edi's theorem for arithmetic progression of arbitrary length. %\\
	
\end{enumerate}

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