Many cliques in bounded-degree hypergraphs

Recently Chase determined the maximum possible number of cliques of size t in a graph on n vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have m edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For s-graphs with s ≥ 3 a number of issues arise that do not appear in the graph case. For instance, for general s-graphs we can assign degrees to any i-subset of the vertex set with 1 ≤ i ≤ s − 1. We establish bounds on the number of t-cliques in an s-graph H with i-degree bounded by Δ in three contexts: H has n vertices; H has m (hyper)edges; and (generalizing the previous case) H has a fixed number p of u-cliques for some u with s ≤ u ≤ t. When Δ is of a special form we characterize the extremal s-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of Füredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case

A localized approach to generalized Tur\'an problems

Generalized Tur\'an problems ask for the maximum number of copies of a graph $H$ in an $n$-vertex, $F$-free graph, denoted by ex$(n,H,F)$. We show how to extend the new, localized approach of Brada\v{c}, Malec, and Tompkins to generalized Tur\'{a}n problems. We weight the copies of $H$ (typically taking $H=K_t$), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of $H$, and in each case prove a tight upper bound on the sum of the weights. A consequence of our new localized theorems is an asymptotic determination of ex$(n,H,K_{1,r})$ for every $H$ having at least one dominating vertex and mex$(m,H,K_{1,r})$ for every $H$ having at least two dominating vertices.Comment: 25 page

After the pit is full: understanding latrine emptying in Fort Dauphin, Madagascar

Faecal sludge management (FSM) remains a challenge for developing countries, particularly in urban areas. This study investigated the barriers to pit latrine emptying in the urban commune of Fort Dauphin, Madagascar through household surveys, focus groups, and key informant interviews. On average, three households were sharing each of the latrines in the study and 20.4% of observed latrines were full. This research established that while no cultural barriers to latrine emptying appear to exist, other challenges include space, finding an emptier, and cost. The rapidity of shared latrine filling, lack of hygienic emptying services, and the absence of faecal sludge disposal or management sites will hinder public health in Fort Dauphin. Affordable access to hygienic emptying and FSM are the forthcoming challenges for sanitation projects in high-density urban communes

The zero forcing polynomial of a graph

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph $G$ of order $n$ as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is the number of zero forcing sets of $G$ of size $i$. We characterize the extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of $\mathcal{Z}(G;x)$, including multiplicativity, unimodality, and uniqueness.Comment: 23 page

Many triangles with few edges

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with n vertices and maximum degree at most r, where n = a(r + 1) + b and 0 ≤ b ≤ r, aKr+1 ∪ Kb has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh and Sudakov conjectured that aKr+1 ∪Kb also maximizes the number of complete subgraphs Kt for each fixed size t ≥3, and proved this for a = 1. Cutler and Radcliffe proved this conjecture for r ≤ 6. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that aKr+1 ∪C(b), where C(b) is the colex graph on b edges, maximizes the number of triangles among graphs with m edges and any fixed maximum degree r ≤ 8, where m = a(r+1 2 ) + b and 0 ≤ b < (r+1 2 ). Mathematics Subject Classifications: 05

Health Care Coverage: Uninsurance -- The Unintended Consequence

One of welfare reform's unintended consequences has been a reduction of health care coverage among poor Americans. The welfare law severed the link between cash assistance and Medicaid. In turn, Congress provided states with several options to continue to offer Medicaid to those leaving welfare and to expand health coverage to more low-income families. Nonetheless, many low-income people lost health care coverage as they moved from welfare to work. This paper provides a statistical portrait of changes in health insurance coverage, and the policy measures that states have taken to fix the problem

The tripartite-circle crossing number of graphs with two small partition classes

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of a tripartite graph is the minimum number of edge crossings among all its tripartite-circle drawings. We determine the exact value of the tripartite-circle crossing number of $K_{a,b,n}$, where $a,b\leq 2$.Comment: 22 pages, 11 figures. Added new results and revised throughout. Originally appeared in arXiv:1910.06963v1, now removed from arXiv:1910.06963v

The Grizzly, September 15, 2016

Board Chair Marcon Resigns Amid Controversy • Meet the Interim Board Chair • Black Girl Dangerous Comes to Speak at Ursinus • Student Work Hits the Stage • A Creative Approach to Raising Awareness • Opinions: Choose the America You Wish to be a Part of ; Students\u27 Guide to Weekends at Reimert • Field Hockey Off to a Hot Start, Looking for Redemption • You Bend \u27Em, We Mend \u27Em: The Life of an Athletic Trainerhttps://digitalcommons.ursinus.edu/grizzlynews/1648/thumbnail.jp

Computer-Enhanced Visual Learning Method to Teach Endoscopic Correction of Vesicoureteral Reflux: An Invitation to Residency Training Programs to Utilize the CEVL Method

Herein we describe a standardized approach to teach endoscopic injection therapy to repair vesicoureteral reflux utilizing the CEVL method, an internet-accessed platform. The content was developed through collaboration of the authors' clinical and computer expertises. This application provides personnel training, examination, and procedure skill documentation through the use of online text with narration, pictures, and video. There is also included feedback and remediation of skill performance and teaching “games.” We propose that such standardized teaching and procedure performance will ultimate in improved surgical results. The electronic nature of communication in this journal is ideal to rapidly disseminate this information and to develop a structure for collaborative research