We owe the T-shirt design and our 2008 contest logo to
W. Stoltzfus of LSU Math Department.
In the study of connected ribbon graphs (that is, connected
graphs, together with an embedding in an oriented surface
with polygonal faces), a quasi-tree is a sub-ribbon graph
which contains a spanning tree and sufficient additional
edges to "minimally carry" the quasi-tree's
Recall that a connected graph with v vertices and e edges
has a spanning tree with the same vertex set and only
v - 1 edges. Similarly, a quasi-tree of a ribbon graph
is a connected, spanning sub-ribbon graph with only one
face and e = v+2 g - 1 edges, for an integer g called
This concept was used by in a paper "Quasi-tree
expansion for the Bollobás-Riordan-Tutte polynomial"
by Oliver T. Dasbach, David Futer, Efstratia Kalfagianni,
Xiao-Song Lin and Neal W.
Stoltzfus to prove that the determinant of a knot is the
alternating sum (with respect to genus) of
the number of quasitrees in a certain ribbon graph constructed
from a knot projection.
For more information and applications, see