List of Student Research Problems in COMAP's Consortium (2023)

Prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

For several years now I have prepared student research problems for COMAP"s newsletter Consortium.

I list here those problems, which issue of Consortium it appeared in and some comments about these problems.

1 (Consortium 112) Folding clusters of equilateral triangles

A polyiamond is a collection of equilateral triangles in the plane where triangles that meet each other share at least an edge. If P is a convex 3-polytope with equilateral triangles with faces than P's graph can be cut along a spanning tree and unfolded to a polyiamond. However, what about folding polyiamonds to non-convex polyhedra to convex polyhedra whose faces are unions of equilateral triangles. This article poses questions related to this circle of ideas.

2. (Consortium 113) Drawings of Number Sequences

The valences of plane graphs obey diophantine equations that can be derived from Euler's polyhedral formula, V +F-E=2. Given positive integers it is interesting to draw graphs with the given valences or degrees.

This problem deals specifically with the integers that can arise as the valences of a simple polygon in the plane that has been subdivided into triangles, something that can always be done, sometimes in many ways.

3. (Consortium 114) The "Middle" of a Graph: Partitions, Eccentricities and Degrees of Vertices in a Graph.

The arithmetic mean and median are examples of measures of central tendency for a data set. For a dots and lines graph one can define a notion of distance between vertices in the graph. Using this concept one can define the "center" of a graph, a set of vertices (sometimes a unique vertex) to which all the others are not too far away.

4. (Consortium 115) Tetrahedra

Given six positive numbers, perhaps 6 integers, when can these numbers serve as the edge lengths of a 3-dimensional convex tetrahedron? Little is known about the 6-dimensional space collection of numbers for which a tetrahedron will exist. Also, 4 points no 3 collinear in the plane give rise to 6 distances which can be be thought of as a degenerate tetrahedron. What can be said about this set of numbers, in particular when they are positive integers?

5. (Consortium 116) Convex Isosceles Triangle Polyhedra

While there are only 8 convex 3-dimensional bounded polyhedra all of whose faces are equilateral triangles (often called the convex deltahedra) there are many open questions when one looks at polyhedra all of whose faces are strictly isosceles triangles or a mixture of equilateral and isosceles triangles.

6. (Consortium 117) From Trees to Polyhedra

If one has a (finite) tree with at least 4 vertices drawn in the plane and no 2-valent vertices, one can construct a 3-polytopal graph (realizable as a convex 3-dimensional bounded polyhedron) by passing a simple closed curve through the 1-valent vertices (leaves) of the tree. T. There are many variant constructions of interest related to these so called Halin graphs.

7. (Consortium 118) Arrangements of Curves

If one has a 4-valent multi-3-gon plane graph its cut-through paths yield simple closed curves. There are many open problems about what lengths of such circuits can occur for graphs in this class of graphs.

8. (Consortium 119) Folding Polyominoes to Boxes

When can a polyomino (1x1x1 squares in the plane that meet edge-to-edge and have no holes fold to form a 1x1xk box, and more generally, and axbxc box or polycube, the 3-dimensional analog of a polyomino.

9. (Consoritum 120) Spanning Trees of n-Dimensional Cubes

The valences of a spanning tree of an n-cube must satisfy a diophantine equation. Can one always find spanning trees that correspond to all of the solutions of this equation. The questions of interest arise since there the equation has the number of 2-valent vertices of the tree with a 0 coefficient.

10. (Consoritum 121) Convex 3-dimensional polyhedra with triangular faces and exactly two edge-lengths

David Eppstein showed that one of the problems disucssed in 6. above has a negative solution - all triangle faced 3-polytopes can't be realized with strictly isosceles triangles. This note treats questions left open by Eppstein and poses some related questions about triangle faced 3-polytopes whose triangles are either isosceles or equilateral.

11. (Consortium 122) Graphs from Rectangles

Given a collection of axis aligned rectangles drawn in the plane one can form a graph from this collection of rectangles by interpreting the places where the edges of the rectangles meet as vertices of a graph, in addition to the vertices of the original rectangles. Various questions related to planar graphs, polyhedral graphs and partitions can be investigated.

11. (Consortium 123) Exploring Games

An introduction to matrix games, in particular 2x2 games and ideas about Nash Equilibria and how to play such games.

11. (Consortium 124) Boxes

Classifying boxes, that is combinatorial 3-cubes when realized metrically, using ideas related to partitions of 12, the number of edges of the cube and 6, the number of faces of a 3-cube. So one can try to classify combinatorial 3-cubes by partition type of edges, and partition type of the number of congruent faces.