Supersaturation Problem for Color-Critical Graphs

The \emph{Tur\'an function} \ex(n,F) of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine $h_F(n,q)$, the minimum number of copies of $F$ that a graph with $n$ vertices and \ex(n,F)+q edges can have. We determine $h_F(n,q)$ asymptotically when $F$ is \emph{color-critical} (that is, $F$ contains an edge whose deletion reduces its chromatic number) and $q=o(n^2)$. Determining the exact value of $h_F(n,q)$ seems rather difficult. For example, let $c_1$ be the limit superior of $q/n$ for which the extremal structures are obtained by adding some $q$ edges to a maximum $F$-free graph. The problem of determining $c_1$ for cliques was a well-known question of Erd\H os that was solved only decades later by Lov\'asz and Simonovits. Here we prove that $c_1>0$ for every {color-critical}~$F$. Our approach also allows us to determine $c_1$ for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge.Comment: 27 pages, 2 figure

Diversity in the Beauty Industry

Because the cosmetics industry tends to set the standard of beauty in society, the lack of representation of racial and/or ethnic minorities in the beauty industry can have psychological impacts on minority consumers. These impacts are to the point where their perceptions of beauty have been skewed to exclude themselves as a result of not seeing themselves represented in advertisement campaigns and product releases by major cosmetics companies. In this lack of representation, consumers’ self-perception, self-esteem, and self-confidence can be negatively influenced. The purpose of this study is to examine the negative impacts that that the lack of minority representation in the cosmetics industry has on specifically LMU students, through short surveys to be distributed online via Qualtrics

Multiple Petersen subdivisions in permutation graphs

A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of M is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen subdivisions in a permutation graph with no 4-cycles, and give a construction showing that this lower bound is tight up to a constant factor

The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $\mathbf{k}$ and $n$, there is a complete multipartite graph $G$ on $n$ vertices with $F(G;\mathbf{k}) = F(n;\mathbf{k})$. Also, for every $\mathbf{k}$ we construct a finite optimisation problem whose maximum is equal to the limit of $\log_2 F(n;\mathbf{k})/{n\choose 2}$ as $n$ tends to infinity. Our final result is a stability theorem for complete multipartite graphs $G$, describing the asymptotic structure of such $G$ with $F(G;\mathbf{k}) = F(n;\mathbf{k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So

Effect of Palm Kernel Cake Based Diet on Libido and Semen Quality of Malin X Santa-Ines Lambs

A study was conducted for six months at the farm of the Department of Animal Science, Universiti Putra Malaysia (UPM), Serdang, Selangor (3' 00' N, 101' 42' E) to investigate the effect of palm kernel cake based diet on serving capacity and semen quality of Malin x Santa-lnes crossbred ram lambs. Twenty Malin x Santa-lnes crossbred ram lambs were randomly assigned into three dietary treatments namely (TI) 60% Palm kernel cake + 40% oil palm frond, (T2) 60% Palm kernel cake + 40% oil palm frond supplemented with 23 mgkg dry matter of molybdenum as ammonium molybdate ((NH4)6M07024.4H20a)n d 600mglkg dry matter of Sulfur as sodium sulphate (Na2S04) and (T3) control; 60% concentrate of corn-soybean mix + 40% oil iii palm frond. There was no significant difference (p>0.05) among the rams for the three dietary treatments for libido or mating behavior. All rams reached their optimum mating behavior (number of ejaculation = 6.7f1.3 and mating efficiency = 0.43kO.01) after the second month of the experiment. It was also found that there was no significant difference (p>0.05) for all the semen quality parameters among the rams under the three dietary treatments. Rams in all dietary treatments produced good semen and their sperm characteristics were within the ranges for good rams (semen volume = 0.7-2.0 ml; sperm concentration =2-5 X lo9 celldml; sperm motility > 80%; percentage of abnormal morphology 4 5%). Although there was a significant difference on the level of copper (Cu) accumulated in the liver (pc0.01) and testis (p<0.05), with highest values in rams under T1 (1089 + 189 ppm and 0.63 f 0.18 ppm, respectively), there was no symptom of toxicity observed in the rams. Therefore, it was concluded that PKC has no negative effect on serving capacity and semen quality of Malin x Santa-lnes crossbred sheep, at least for six months of the feeding period. It was also concluded that supplementing 23 mglkg dry matter of Molybdenum and 600 mgkg dry matter Sulphur to sheep fed on PKC based diet helps to protect Cu accumulation in the plasma, liver and testis of rams without negative effect on their reproductive capacity and semen quality. Even though higher serum testosterone was found in rams under diets T1 and T2 compared to T3, this did not reflect the actual reproductive level and sperm quality. Hence, it was concluded that serum testosterone level might not be a good indicator of reproductive performance of rams when blood samples are taken from conscious animals

EAP alignment in EFL writing courses towards developing student’s academic writing skills

The objective of the present study was to assess the prevailing alignment of English for academic purposes (EAP) in EFL writing courses vis-à-vis the development of academic writing skills of students in public universities of Amhara National Regional State (ANRS) in Ethiopia. The study used a descriptive survey that involved quantitative and qualitative methods. A questionnaire, focus-group discussions, and syllabus and materials survey were used to gather primary and secondary data, respectively, for the study. The descriptive statistics were employed in analyzing the quantitative data. The qualitative data were analyzed and reported thematically. The results showed there were existed a comparable horizontal (cross-department) distribution pattern of common writing courses among the three universities; all learners in taking the common course 'Basic Writing Skills' in each university learnt with the same syllabus and material regardless of their fields of study. So, there was a poor alignment of writing courses with the target students' mainstream discipline (i.e., learning and discourse) both in terms of content and genre.

A new bound for the 2/3 conjecture

We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value, proved by Erdos, Faudree, Gould, Gy\'arf\'as, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice

Relationship between Working Capital Management, Policies, and Profitability of Small Manufacturing Firms

Working capital optimization, as an act of balancing liquidity and profitability, presents significant challenges when small businesses lack managerial expertise and access to affordable capital and credit facilities. To remain successful through efficient utilization of working capital, small business leaders need to understand the association between working capital management (WCM), working capital policy (WCP), and business profitability (PFT). Anchored in the cash conversion cycle theory, the purpose of this correlational study was to examine the relationship between WCM, WCP, and PFT. The study employed a retrospective secondary analysis of financial data from 2004 to 2013 from a random sample of 176 publicly traded small U.S. manufacturing companies. The regression results incorporating 3 models were significant in predicting profitability in terms of gross operating profit (GOP), return on asset (ROA), and Tobin\u27s q (TBQ). The regression results showed that WCM and WCP were significant predictors of GOP, F (5, 170) = 8.580, p \u3c .000, R2 = .201; ROA, F (5, 170) = 4.079, p \u3c .002, R2 = .107; and TBQ, F (5, 170) = 6.231, p \u3c .000, R2 = .155. The overall result confirmed that WCM and WCP predicted PFT significantly (p \u3c .05). Small business leaders may incorporate working capital optimization practices into overall corporate strategy, thereby aligning working capital needs with the changing business requirements. The implications for positive social change included the potential to provide small business leaders with knowledge of WCM and WCP as drivers of PFT. Profitable businesses may provide employees and communities with better jobs; stock ownership; and development infrastructures such as road, healthcare, and educational facilities

Community Attitudes Toward Wildlife and Protected Areas in Ethiopia

Across Africa, national policies that established protected areas (PAs) typically limited local use of wildlife and other resources. Over time, these policies have raised tensions with rural communities and today threaten to undermine conservation goals. This article examines community–PA relationships at four important sites in Ethiopia—a country of rich tradition with an unusual colonial past. Using focus groups and household surveys, we found that despite local tensions, most respondents held positive views toward wildlife and nearby PAs. Factors influencing positive views included receiving PA benefits, good relations with PA staff, higher education levels, being older, having a large family, diversified income sources, owning fewer livestock, and fewer incidents of wildlife conflicts. In contrast, the devolved control of PAs from federal to regional levels has not influenced community–PA relations as intended. Our results suggest that relations could be improved through involving communities in co-management arrangements, honoring resource tenure and use rights, providing benefits, and implementing conservation education programs