Inside-Out Polytopes

We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat

Friendly index sets of starlike graphs

For a graph G = (V, E) and a coloring (labeling) f : V(G) → Z2 let vf(i) = | f-1(i)|. The coloring f is said to be friendly if |vf(1) - v f(0)| ≤ 1. The coloring f : V( G) → Z2 induces an edge labeling f* : E( G) → Z2 defined by f* (xy) = f( x) + f(y) (mod 2). Let ef(i) = |f*-1( i)|. The friendly index set of the graph G, denoted by FI (G), is defined by FIG= ef1-ef 0:f isafriendly vertexlabelingof G. In this thesis the friendly index sets of certain classes of trees, called starlike graphs, will be determined

Testing Convexity and Acyclicity, and New Constructions for Dense Graph Embeddings

Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree. Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces