For 2016, I plan to have my group study another version of the dimension of a graph. I am optimistic that this version will be interesting and take up most of the summer. As a side project, we might also study a certain topological property of graphs embedded in the projective plane.
There is no specific prerequisite background for working in my group, aside from strong mathematical maturity. I will be looking for students who are ready to follow up on their work beyond the summer program.
Joel Foisy's REU group history:
2015: The group started with a classic paper, "On the dimension of a graph," by Erdos, Harary and Tutte and then worked on some related problems. Perhaps a general classification for these problems would be "geometric graph theory." They wrote a paper, currently under revision, studying the dimension of a graph, with a definition of dimension similar to, but different from, the definition of Erdos, Harary and Tutte.
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. (Recently accepted for publication.)
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, before my hair turned grey:
Earlier that same day at nearby Owl's Head Mountain: