In 2015, my plan is to start with a classic paper, "On the dimension of a graph," by Erdos, Harary and Tutte and then work on some related problems. Perhaps a general classification for these problems would be "geometric graph theory." We might also study some topological graph theory problems.
There is no specific prerequisite background for working in my group, aside from strong mathematical maturity.
Joel Foisy's REU group history:
2014: The group studied three topics and wrote a whopping 3 papers. One has been submitted already (good job Madeleine!) The other two are in progress. Here are links to them: Intrinsically spherical linked graphs and Linking of n-spheres.
2013: We ended up studying a purely graph theoretic idea: weakly diameter m critical graphs. The main paper is going to appear in Graph Theory Notes of New York: On weakly diameter-m-critical graphs,
2012: Two topics. (Papers are still being edited, the first one has some deep issues that I hope will soon be resolved, the second is in the hands of one of the student authors and hopefully will be submitted very soon): the generalized conflict graph,
2010: Flexibly planar and flexibly flat graphs.
2008: Intrinsically 3-linked graphs in projective space.
2006: Lower dimensional versions of intrinsically linked graphs. (Paper published in Rose Hulman Journal: Intrinsically S^1 3-linked graphs an other aspects of S^1 embeddings).
2003: Knotted Hamiltionian cycles in spatial graphs. (Two papers published: Some results on intrinsically knotted graphs , with Tom Fleming and REU students Garry Bowlin, Paul Blain, Jacob Hendricks, and Jason LaCombe,J. Knot Theory Ramif., (16), no. 6 (2007); 749-760. Also: Knotted Hamiltonian cycles in spatial embeddings of complete graphs , with former REU '03 students Garry Bowlin, Paul Blain, Jacob Hendricks, and Jason LaCombe, New York J. Math. (13), (2007); 11-16.)
2002: Intrinsic Chirality. Their results were already known, as we learned later. They did complete a short paper, but I do not have it available electronically.
2001: Intrinsically linked graphs with an unused vertex.
1999: Disjoint Linking property of spatial graphs. Their results were combined with the 2000 group's results in the paper listed above.
1997: Perimeter minimizing enclosures, using a corner. (Paper published: J. Foisy, G. Christopher Hruska, Dmitriy Leykekhman, Daniel Pinzon, and Brian J. Shay; Rocky Mountain J. of Math., (31), no. 2 (2001); 437-482.)
You might want to check out:
Some participants in our 2003 program in the Adirondacks:
Earlier that same day at nearby Owl's Head Mountain: