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 automatic virtual environment at Supercomputing'95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.

We consider the subchromatic number χ S (G) of graph G, which is the minimum order of all partitions of V (G) with the property that each class in the partition induces a disjoint union of cliques. Here we establish several bounds on subchromatic number. For example, we consider the maximum subchromatic number of all graphs of order n and in so doing answer a question posed in [20]. We also consider bounds on χ S (G) when the size and genus of G are known. We also consider the parameter when applied to planar and outerplanar graphs. It is known that the problem of determining whether χ S (G) ≤ k is NP-complete for all k ≥ 2. We extend this by showing it is NP-complete for k = 2 even when restricted to the class of planar triangle-free graphs with maximum degree four. As a corollary we see that showing a planar triangle-free graph of maximum degree four has a 1-defective chromatic number of two is NP-complete, answering a question of [8]. We show that determining whether χ S (G) ≤ 3 is NP-complete for planar graphs. We consider the subchromatic number of cartesian products of complete graphs and show a correspondence with a natural covering of matrices. We close by producing bounds on the subchromatic number in terms of chromatic number as well as the product of clique number with chromatic number. Sharpness for graphs with fixed clique size is discussed.

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Abstract. We consider several tools for computing and visualizing sphere eversions. First, we discuss a family of rotationally symmetric eversions driven computationally by minimizing the Willmore bending energy. Next, we describe programs to compute and display the double locus of an immersed surface and to track this along a homotopy. Finally, we consider ways to implement computationally the various eversions originally drawn by hand; this requires interpolation of splined curves in time and space.

An obstacle representation of a graph G is a straight-line drawing of G in the plane together with a collection of connected subsets of the plane, called obstacles, that block all non-edges of G while not blocking any of the edges of G. The obstacle number obs(G) is the minimum number of obstacles required to represent G.We study the structure of graphs with obstacle number greater than one. We show that the icosahedron has obstacle number 2, thus answering a question of Alpert, Koch, & Laison asking whether all planar graphs have obstacle number at most 1. We also show that the 1-skeleton of a related polyhedron, the gyroelongated 4-bipyramid, has obstacle number 2. The order of this graph is 10, which is also the order of the smallest known graph with obstacle number 2.Some of our methods involve instances of the Satisfiability problem; we make use of various "SAT solvers" in order to produce computer-assisted proofs.