A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic $k$-coloring can be refined to a star coloring with at most $(2k^2-k)$ colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width $t$ can be star colored with ${t+2\choose2}$ colors, and we show that this is best possible.

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.

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.

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Let G be an n-vertex graph with list-chromatic number χ ℓ . Suppose each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas [1] conjecture that at least tn χ ℓ 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 6 7 of the conjectured number can be colored.