I am a participant of this year's REU program. We work on several problems in graph theory.
Hamiltonian cycles Let consider connected graphs with at least two vertices.
The proof will appear here soon. Shannon capacity is a interesting parameter of a graph, it is extremely difficult to compute. This would by very interesting, since this would be the very first example of a graph, for which the Shannon capacity is the third root of an integer. Suppose you have the edges of complete graph on N vertices by two colors. There is also known a construction in which you cannot find any larger complete graph.