Approximate Algorithm for Finding maximum upward emeddable sub - digraph of an acyclic

Graphs appear as simplified representations of a diversity of real-world structures. Naturally there is a question of an intrinsic graph metric, often presumed to be the “shortestpath” metric. Other plausible possibilities for intrinsic graph metrics are formulated in terms of the (combinatorial) Laplacian matrix of the graph, as well as in terms of other combinatoric, probabilistic, and physical frameworks. The physically motivated candidates are developed in terms of electrical resistances and wave-amplitude correlations. Some theorems concerning the consequent candidate metrics are noted. Granted an intrinsic metric, there are consequent graph invariants. Two natural “graph cyclicity” invariants to measuring the degree of cyclicity of a graph are noted, along with some associated theorems. Further invariants may be defined through analogy to quantities defined for Euclidean geometries. Such include: linear curvature, torsion, Gaussian curvature, and a sequence of volumina measures. One might surmise that there arises the possibility of some sort of “graph geometry”. Contributed Talks IIIA, Saturday 2:20-3:30pm 2:202:40 Jozsef Balogh, Ohio State University. Joint with P. Keevash and B. Sudakov. Title: Disjoint Representability of sets. Abstract: For a hypergraph H and a set S, the trace of H on S is the set of all intersections of edges of H with S. We will consider forbidden trace problems, in which we want to find the largest hypergraph H that does not contain some list of forbidden configurations as traces, possibly with some restriction on the number of vertices or the size of the edges in H. Write [k](`) for the set of all `-subsets of [k] = {1, · · · , k}. Note that A has k disjointly representable sets exactly when it has a [k](1) trace. We will focus on three forbidden configurations: the k-singleton [k](1), the k-co-singleton [k](k−1) and the k-chain Ck = {∅, {1}, [1, 2], · · · , [1, k − 1]}. We prove a number of results on the size of the largest hypergraph H with some combination of these traces forbidden, sometimes with restrictions on the number of vertices or the size of the edges. We obtain exact results in the case k = 3, both for uniform and non-uniform hypergraphs, and classify the extremal examples, and asymptotical results for larger values of k. This is joint work with P. Keevash and B. Sudakov. 2:453:05 Rong Luo, Middle Tennessee State University. Title: Coloring edges of graphs embedded in a surface of characteristic −3. Abstract: Consider graphs that are embeddable in a surface of characteristic −3. It is known that class two graphs of this type with maximum degree at most 8 exist. Yan and Zhao showed that such graphs with maximum degree at least 10 must be class one. In this talk, we show that such graphs with maximum degree 9 also must be class one, completing the analysis of these surfaces.