coloring algorithms, dichromatic polynomial
Eulerian / Hamiltonian paths
given 2 faces, tell distance for pieces like those of type C (like a Chess knight), and given a start and end direction, construct the shortest (geodesic) path (that is element of the 'vector' group) as well (is it well-defined and possible in polynomial time?) according to variational principle (smallest Wirkung, least action)
how a Chess knight(2,1) on some torus(m,n) sees the world (infile with poly=8 to get the equivalent polyhedron for a piece of type A)
inverse problem: give matrices or polynomials and recontruct the graph (several graphs in case there is no unique solution)
verify the Kuratowski theorem: any non-planar graph contains K5 or K3,3
what about the chain of polynomials constructed by using a graph with boundary, and then taking duals successively, thereby reducing the graph step by step (after 2 steps the inner boundary lines are gone, replaced by their neighbour lines, not necessarily meaning that always faces are removed because they may be shared)
when making a face invalid how to achieve the resulting char. polynom from the original intact surface' polynom?
optionally take holes as valid faces in polynomials creation
take the game graph (positions are the vertices with moves as edges) and create characteristic polynomials etc.
in order to be able to construct groups and polynomials for graphs with faces connected over multiple edges (see the Rubik cube 3x3x3 after 2x2 rotation and hexa_unorient) modify the used methods accordingly (the adjacency matrix entries are the number of connecting edges, so instead of only 0 or 1 now there other integers like 2 are allowed)
for the unorientable cases: what happens when we identify the faces from the two orientations, do we get complex roots in the corresponding polynomials (see hexa_unorient and proj_plane)?
are matroids involved in our project?