Copyright (c) 2013 Utah State University All rights reserved. http://digitalcommons.usu.edu/mathsci_presentations Recent documents in Mathematics and Statistics Faculty Presentations en-us Wed, 20 Mar 2013 01:46:45 PDT 3600 http://digitalcommons.usu.edu/mathsci_presentations/53 http://digitalcommons.usu.edu/mathsci_presentations/53 Mon, 18 Mar 2013 16:00:55 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/54 http://digitalcommons.usu.edu/mathsci_presentations/54 Mon, 18 Mar 2013 16:00:55 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/51 http://digitalcommons.usu.edu/mathsci_presentations/51 Mon, 18 Mar 2013 16:00:54 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/52 http://digitalcommons.usu.edu/mathsci_presentations/52 Mon, 18 Mar 2013 16:00:54 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/50 http://digitalcommons.usu.edu/mathsci_presentations/50 Mon, 18 Mar 2013 16:00:53 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/49 http://digitalcommons.usu.edu/mathsci_presentations/49 Mon, 18 Mar 2013 16:00:52 PDT We introduce a series of generalizations of probe interval graphs called t-probe interval graphs, (a probe interval graph is a 1-probe interval graph) and show, via a method similar to graph homomorphism, that each class, including the class of probe interval graphs, is contained in the class of interval k-graphs. Any probe interval graph is clearly a tolerance graph, but for some t>1 this relationship fails. We wish to determine this t. Also, the interval k-graphs whose complement describes a poset are believed to have a nice characterication via forbidden subgraphs, and we give the conjecture here, and a new description of these interval k-graphs that is similar to the salient property of function graphs.

]]> David E. Brown et al. http://digitalcommons.usu.edu/mathsci_presentations/48 http://digitalcommons.usu.edu/mathsci_presentations/48 Mon, 18 Mar 2013 16:00:51 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/46 http://digitalcommons.usu.edu/mathsci_presentations/46 Mon, 18 Mar 2013 16:00:50 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/47 http://digitalcommons.usu.edu/mathsci_presentations/47 Mon, 18 Mar 2013 16:00:50 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/44 http://digitalcommons.usu.edu/mathsci_presentations/44 Mon, 18 Mar 2013 16:00:49 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/45 http://digitalcommons.usu.edu/mathsci_presentations/45 Mon, 18 Mar 2013 16:00:49 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/42 http://digitalcommons.usu.edu/mathsci_presentations/42 Mon, 18 Mar 2013 16:00:48 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/43 http://digitalcommons.usu.edu/mathsci_presentations/43 Mon, 18 Mar 2013 16:00:48 PDT A graph G is a tolerance graph if and only if each vertex v ∈ V (G) can be associated with an interval Iv of the real numbers and a positive real number tv with uv ∈ E(G) if and only if |Iv ∩ Iu | ≥ min{tv , tu }. Graph G is a probe interval graph if there is a partition of V (G) into sets P and N with each vertex associated to an interval of the real number line such that uv ∈ E if and only if Iu ∩ Iv ̸= Ø and {u, v} ∩ P ̸= Ø. We give a recognition algorithm for bipartite tolerance graphs that yields a structural characterization in terms of 2-connected blocks. With a few modifications, the same recognition algorithm works for bipartite probe interval graphs and yields a structural characterization for them in terms of 2-edge-connected blocks. The recognition algorithm is O(|V | + |E|) for both classes of graphs.

]]> David E. Brown et al. http://digitalcommons.usu.edu/mathsci_presentations/41 http://digitalcommons.usu.edu/mathsci_presentations/41 Mon, 18 Mar 2013 16:00:47 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/39 http://digitalcommons.usu.edu/mathsci_presentations/39 Mon, 18 Mar 2013 16:00:46 PDT We present an easy-to-implement 3-axis rubric for the formative and summative assessment of open-ended solutions and proofs. The rubric was constructed for the use on the written work of students in a Discrete Mathematics class at a research-oriented university, with the following in mind: (1) To aid in the efficiency and consistency of assessment of proofs and open-ended solutions, with the possibility of being comfortably implemented by an undergraduate assistant; (2) To provide the simultaneous formative and summative assessment of the students’ written work. Thus, the questions we address are: How can we foster good technical writing skills in a way that improvement can be measured? How can large amounts of written work be processed and assessed so that summative and formative judgments are passed but without much time used by the instructor/professor/TA? The axes we use are labeled validity, readability, and fluency, corresponding to (respectively) correctness of calculations and deductions, the ease with which the solution or proof can be read, and the extent to which a student is able to use and communicate via the technical notions relevant to the problem or proof — for example appropriateness and correctness of notation. The rubric format is communicated to the students and discussed in class before any written work is assessed. The rubric has been implemented by professors and teaching assistants only after being trained in its use.

]]> David E. Brown et al. http://digitalcommons.usu.edu/mathsci_presentations/40 http://digitalcommons.usu.edu/mathsci_presentations/40 Mon, 18 Mar 2013 16:00:46 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/38 http://digitalcommons.usu.edu/mathsci_presentations/38 Mon, 18 Mar 2013 16:00:45 PDT If an interval graph is such that its complement can be oriented transitively, that orientation yields an interval order. A graph G is an interval-probe graph if its vertices can be partitioned into P (probes) and N (nonprobes) and each vertex v can correspond to an interval Iv so that vertices u and v are adjacent if and only if Iu ∩Iv ̸= Ø and {u,v}∩P ̸= Ø. A graph G is an interval k-graph if its vertices can be properly colored and each vertex v can correspond to an interval Iv so that vertices u and v are adjacent if and only if Iu ∩Iv ̸= Ø and u and v are colored differently. Interval probe graphs generalize interval graphs and interval k-graphs generalize interval-probe graphs. This talk will contain recent characterizations of interval- probe orders (order obtained from a transitive orientation of an interval-probe graph) of interval k-orders (order obtained from a transitive orientation of an interval k-graph).

]]> David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/36 http://digitalcommons.usu.edu/mathsci_presentations/36 Mon, 18 Mar 2013 16:00:44 PDT In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. In this talk, we discuss some preliminary results on a generalization of the concept of cycle- extendability to S-extendable; that is, with S ⊆ {1, 2, . . . , n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i ∈ S. We present some results on tournaments, i.e., complete directed graphs, and some observations about cycle-extendability and S-extendability for non-directed graphs.

]]> LeRoy B. Beasley et al. http://digitalcommons.usu.edu/mathsci_presentations/37 http://digitalcommons.usu.edu/mathsci_presentations/37 Mon, 18 Mar 2013 16:00:44 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/34 http://digitalcommons.usu.edu/mathsci_presentations/34 Mon, 18 Mar 2013 16:00:43 PDT David E. Brown http://digitalcommons.usu.edu/mathsci_presentations/35 http://digitalcommons.usu.edu/mathsci_presentations/35 Mon, 18 Mar 2013 16:00:43 PDT David E. Brown