Glenn G. Chappell

The minimax sphere eversion

We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin’s classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE TM automatic virtual environment at Supercomputing’95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.

A Matroid Generalization of a Result on Row-Latin Rectangles

Let A be an m×n matrix in which the entries of each row are all distinct. A. A. Drisko (1998, J. Combin. Theory Ser. A84, 181?195) showed that if m?2n?1, then A has a transversal: a set of n distinct entries with no two in the same row or column. We generalize this to matrices with entries in the ground set of a matroid. For such a matrix A, we show that if each row of A forms an independent set, then we can require the transversal to be independent as well. We determine the complexity of an algorithm based on the proof of this result. Finally, we observe that m?2n?1 appears to force the existence of not merely one but many transversals. We discuss a number of conjectures related to this observation (some of which involve matroids and some of which do not).

A lower bound for partial list colorings

Let G be an n-vertex graph with list-chromatic number Cl. Suppose that each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas [1] conjecture that at least tn-Cl vertices can be colored from these lists. We prove a lower bound for the number of colorable vertices. As a corollary, we show that at least

Thresholds for Path Colorings of Planar Graphs

{6}\over{7}

Uniquely Tree-saturated Graphs

of the conjectured number can be colored.

Interactive voxel graphics in virtual reality

A graph is path k-colorable if it has a vertex k-coloring in which the subgraph induced by each color class is a disjoint union of paths. A graph is path k-choosable if, whenever each vertex is assigned a list of k colors, such a coloring exists in which each vertex receives a color from its list.

Polyunsaturated posets and graphs and the Greene-Kleitman theorem

Let H be a graph. A graph G is uniquely H-saturated if G contains no subgraph isomorphic to H, but for every edge e in the complement of G (i.e., for each “nonedge” of G),

Bounds on the metric and partition dimensions of a graph

G+e