Deepak Bal (Montclair State University)

Title: Rainbow Structures In Random Graphs

Abstract: Consider a graph whose edges have been assigned colors. A set of edges is called rainbow if all the edges in the set receive different colors. In this talk I will discuss some results concerning the existence rainbow structures (matchings, Hamilton cycles, spanning trees) in various models of random graphs (Erdös - Rényi, random directed, random geometric) when the edges have been assigned colors randomly.

Sandra Kingan (Brooklyn College)

Title: Graph Generation Using The Strong Splitter Theorem

Abstract: The Splitter Theorem for graphs states that, if H is a simple 3-connected proper minor of a simple 3-connected graph G such that, if H is a wheel then G has no larger wheel, then there exists a sequence G₁ ... G_t of simple 3-connected graphs with G₁ isomorphic to H, and G_t=G such that, for i in {2, ... t}, G_i is a simple single-edge extension or cosimple single-edge coextension of G_i-1. The Strong Splitter Theorem optimizes the Splitter Theorem to best possible by showing that we can obtain up to isomorphism G starting with H and at each step doing a simple single-edge extension or cosimple single-edge coextension, such that at most two consecutive single-edge extensions may occur in the sequence before a single-edge coextension must occur, unless the rank of the minors involved are the same as the rank of G. In this talk we present an algorithm for 3-connected graph generation using the Strong Splitter Theorem. A portion of this talk is joint work with Manoel Lemos.

Joseph Malkaevitch (York College)

Title: Euler Relation Insights For 3-Polytopes Having Hamiltonian Circuits

Abstract: Euler's polyhedral formula - V + F - E = 2 - for convex 3-polytopes and plane connected graphs continues to generate new insights and questions about the structure of plane graphs. This talk examines some new results about Hamiltonian circuits of special classes of 3-polytopes using "Euler relations" and shows how these results grow out of past work.

Ariane Masuda and Satyanand Singh (New York City College of Technology)

Title: Extended Calkin-Wilf Trees

Abstract: The Calkin Wilf tree can be traced back to Kepler while he studied planetary motion. We will derive symmetries and asymmetry in a generalization of the Calkin Wilf tree due to Nathanson. We will also show how these Calkin Wilf generalizations and their associated matrices allow us to further characterize these trees which have important applications in discrete mathematics and number theory. This is joint work with Sandie Han, Ariane Masuda, Satyanand Singh and Johann Thiel.

David Offner (Westminster College)

Title: Polychromatic Colorings Of The Hypercube

Abstract: Given a graph G which is a subgraph of the n-dimensional hypercube Q_n, an edge coloring of Q_n such that every copy of G contains every color is called G-polychromatic. Denote by p(G) the maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube. The problem of determining p(G) was first studied by Alon, Krech and Szabó in 2007 as a way to prove bounds for Turán type problems on the hypercube. This talk will survey what is currently known about polychromatic colorings on the hypercube and introduce some generalizations and open problems. We will introduce techniques that allow us to determine exact values for p(Q_d) for all d, and prove bounds on p(G) for other subgraphs G of Q_n. We will also discuss generalizations where subcubes of Q_n are colored. This is joint work with Maria Axenovich, John Goldwasser, Bernard Lidicky, Ryan Martin, John Talbot, and Michael Young.

Lakzian Sajjad (Fordham University) and Zachary McGuirk (Hunter College)

Title: Conical Curvature-Dimension Condition With Applications To Spectral Graph Theory

Abstract: In this talk we will introduce a non-trivial notion of curvature for graphs via an inequality for second-order PDEs on the graph's function space. Using a restricted version of this curvature notion for just the cone point over the vertex set we are able to derive a global Poincaré inequality which depends only on the underlying graph and use it to find bounds for the first non-trivial eigenvalue of the graph Laplacian on the underlying graph.

Charlie Suffel (Stevens Institute of Technology)

Title: Domination – An Exceptional Invariant

Abstract: The domination number of a graph is the minimum order of a set of nodes with the property that each node of the graph is either in the set or is adjacent to a node in the set. This parameter is referred to as an exceptional invariant since , in general , removal of a set of nodes from the graph can cause the domination number to increase and removal of another set can cause a decrease in the domination number. We study this phenomenon , indicate some anomalies and prove an interesting result for trees in complete detail.

Micahel Yatauro (Penn State Brandywine)

Title: A Closure Theorem For Graph Tenacity

Abstract: Let G be a finite, simple graph. For a subset S of V(G), let w(G-S) be the number of components of the induced subgraph G-S and let m(G-S) be the order of a largest component of the induced subgraph G-S. The tenacity of G, denoted T(G), is the minimum value obtained by the ratio (|S|+m(G-S)) / w (G-S) when considering all possible subsets S of V(G). If T(G) t, we say that G is t-tenacious. We will prove a theorem which provides a number k so that if deg(x)+deg(y) ³ k for some non-adjacent vertices x, y of G, then G is t-tenacious if and only if G+xy is t-tenacious. It is also shown that our value of k is best possible.