Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Phone: 718-262-2551 (Voicemail available)
web page: http://www.york.cuny.edu/~malk
Understanding the interface between combinatorial and metrical properties of convex 3-dimensional polyhedra
Specific sample investigations:
1. If P is a convex 3-dimensional polyhedron all of whose faces are triangles is there a P* combinatorially equivalent to P all of whose faces are isosceles triangles?
2. Understanding the structure of the:
a. convex 3-dimensional polyhedra all of whose faces are congruent isosceles triangles.
b. convex 3-dimensional polyhedra whose faces are triangles of with side lengths a, a, b and b, b, a.
c. convex 3-dimensional polyhedra whose faces are triangles with two edge lengths.
3. If a 3-polytope P can be realized with triangular faces which are isosceles then the associated planar 3-connected graph G, admits edge colorings of a special kind. There are many interesting graph theory questions about "isosceles colorings" and other types of non-standard edge colorings that can be investigated.
4. I have developed a family of new "Euler relations" which make it possible to get new insights in the 3-polytopal graphs. There are a variety of new questions which grows out of this work. These are similar in spirit to what have come to be called "Eberhard type" problems in the theory of 3-polytopal graphs.
5. Graph theory problems associated with triangulated polygons.
Basic Euclidean geometry, graph theory and combinatorics
Titles of doctoral theses that I have supervised at the Graduate Center:
Some properties of 3-polytopal graphs
Polytopal graphs and arrangements of curves
Spanning trees of Three Polytopal Graphs
Integer sequences associated with trees
Mode of interacting with students:
I have a heavy teaching schedule at York. Ideally I like to meet once a week (for between 1 and 2 hours) at York with my thesis student. If this will not work, I try to find some alternate arrangement. I attend the Geometry Seminar at the Courant Institute regularly so meetings at Courant are sometimes possible, and I get to the Graduate Center on an intermittent basis.