Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.

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Two edge colorings of a graph are edge-Kempe equivalent if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar bipartite cubic graphs with a range of numbers of edge-Kempe equivalence classes. Techniques are developed that will be useful for analyzing other classes of graphs as well.

Abstract:We prove that if G is a triangulation of the torus and (G) = 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors. ᭧