Decomposition of Certain Complete Graphs and Complete Multipartite Graphs into Almost-bipartite Graphs and Bipartite Graphs

In his classical paper [14], Rosa introduced a hierarchical series of labelings called ρ, σ, β and α labeling as a tool to settle Ringel’s Conjecture which states that if T is any tree with m edges then the complete graph K2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco et al. [6] introduced γ-labeling as a stronger version of ρ-labeling. A function g defined on the vertex set of a graph G with n edges is called a γ-labeling if (i) g is a ρ-labeling of G, (ii) G is a tripartite graph with vertex tripartition (A, B, C) with C = {c} and ¯b ∈ B such that {¯b, c} is the unique edge joining an element of B to c, (iii) g(a) \u3c g(v) for every edge {a, v} ∈ E(G) where a ∈ A, (iv) g(c) - g(¯b) = n. Further, Blinco et al. [6] proved a significant result that the complete graph K2cn+1 can be cyclically decomposed into c(2cn + 1) copies of any γ-labeled graph with n edges, where c is any positive integer. Recently, in 2013, Anita Pasotti [4] introduced a generalisation of graceful labeling called d-divisible graceful labeling as a tool to obtain cyclic G-decompositions in complete multipartite graphs. Let G be a graph of size e = d . m. A d-divisible graceful labeling of the graph G is an injective function g : V (G) → {0, 1, 2, . . . , d(m + 1) - 1} such that {|g(u) - g(v)|/{u, v} ∈ E(G)} = {1, 2, . . . , d(m + 1) - 1}\{m + 1, 2(m + 1), . . . , (d - 1)(m + 1)}. A d-divisible graceful labeling of a bipartite graph G is called as a d-divisible α-labeling of G if the maximum value of one of the two bipartite sets is less than the minimum value of the other one. Further, Anita Pasotti [4] proved a significant result that the complete multipartite graph K (e/d +1)×2dc can be cyclically decomposed into copies of d-divisible α-labeled graph G, where e is the size of the graph G and c is any positive integer (K (e/d +1)×2dc contains e/d + 1 parts each of size 2dc). Motivated by the results of Blinco et al. [6] and Anita Pasotti [4], in this paper we prove the following results. i) For t ≥ 2, disjoint union of t copies of the complete bipartite graph Km,n, where m≥ 3, n ≥ 4 plus an edge admits γ-labeling. ii) For t ≥ 2, t-levels shadow graph of the path Pdn+1 admits d-divisible α-labeling for any admissible d and n ≥ 1. Further, we discuss related open problems

Total Colouring of New Classes of Subcubic graphs

The total chromatic number of a graph $G$, denoted $\chi^{\prime\prime}(G)$, is the least number of colours needed to colour the vertices and the edges of $G$ such that no incident or adjacent elements (vertices or edges) receive the same colour. The popular Total Colouring Conjecture (TCC) posed by Behzad states that, for every simple graph $G$, $\chi^{\prime\prime}(G) \leq \Delta(G)+2$. In this paper, we prove that the total chromatic number for a family of subcubic graphs called cube connected paths and also for a class of subcubic graphs having the property that the vertices are covered by independent triangles are exactly $\Delta(G)+1$. More precisely, these two families of subcubic graphs are shown to be Type 1 graph.\

Social Media's Effect on Millennials and Generation Z's Green Purchasing Habits

Purpose: The purpose of researching the impact of social media on millennials' and Generation Z's green purchasing habits is to better understand how these platforms influence their attitudes and behaviours towards environmentally conscious consumer choices. This data can help businesses and marketers target and engage these demographic groups more effectively, as well as shed light on the role of social media in promoting sustainable consumption and driving positive environmental change.   Theoretical framework: There are two theories that can be used to understand the impact of social media on the green purchasing habits of millennials and Generation Z. Social Learning Theory suggests that individuals learn and adopt new behaviors by observing the actions and attitudes of others, including those they follow on social media and Norm Activation Theory states that Individuals are much more likely to engage in environmentally responsible behaviours when they believe such behaviours are the social norm and are supported by their peers and social media influencers.   Design/methodology/approach: A descriptive research design was used to investigate social media's impact on the green purchasing habits of millennials and Generation Z. The current study has conducted a quantitative analysis of the data that was gathered using a questionnaire survey. Based on the age category, the respondents were chosen using the purposive sample technique. Focusing on a particular age group that falls within the millennial and generation Z categories was done via non-probability purposive sampling. There were 600 respondents in the sample used for the study.   Findings: According to the study's findings, social media has a substantial impact on shaping the attitudes and behaviours of generation y and z groups towards environmentally conscious consumer choices.   Research, Practical & Social implications: Businesses and marketers can use the findings of this research to better target and engage these demographic groups in promoting environmentally conscious consumer choices. We can gain a better understanding of the role of social media in promoting sustainable consumption and the potential for these platforms to drive positive change towards more environmentally responsible behaviours by investigating the impact of social media on the green purchasing habits of generation y and z groups.   Originality/value: The originality and value of this study investigating the impact of social media on millennials' and Generation Z's green purchasing habits derive from its potential to inform strategies for promoting sustainable consumption and driving positive change towards a more environmentally responsible future

On hydrodynamic limits in Sinai-type random environments

We investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common `Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a `rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe formal connections through a two scale limit.Comment: 30 page

Morphology Studies of Iron-Manganese Thin Films

Fe-based catalysts are known to be effective for Fischer-Tropsch synthesis from coal but are sensitive to sulfur poisoning. Addition of manganese to these catalysts has been proposed in an effort to combat this catalyst deactivation. To investigate the fundamental physical aspects of Mn incorporation into Fe, different compositions of model thin films of Fe-Mn, ranging from 100% Fe to 100% Mn were studied for the very first time, using scanning tunneling microscopy (STM), atomic force microscopy (AFM) and scanning electron microscopy. Our preliminary results indicate that the grain size of iron varied from 50 nm to 150 nm using the AFM image, in general agreement with others. Also the mixture with small amounts of manganese had large oxidic area. The surface was modified by the addition of manganese. The STM images of the samples showed similarities between pure iron and 75% Fe-25% Mn, while the 25% Fe-75% Mn looked very similar to 100% Mn. The mixture of 50% Fe-50% Mn seemed to contain two distinct surface features. This leads us to believe that STM and AFM can be useful tools to further investigate the model iron-manganese thin films and hence give insight into the unknown atomic scale interactions and also to study the catalytic properties better

Bicycle handlebar injuries - a systematic review of pediatric chest and abdominal injuries.

OBJECTIVE: The severity of handlebar injuries can be overlooked due to subtle signs and wide range of associated internal injuries. Our objective was to describe thoracoabdominal injuries due to bicycle handlebars and their outcomes in children. METHODS: Articles that reported thoracoabdominal injuries were identified from database conception to March 3, 2019 using PubMed, EMBASE, Cochrane Library, CINHAHL Complete, Web of Science and Scopus. A systematic review of studies of thoracoabdominal handlebar injuries in children ≤21 years on human-powered bicycles in English was performed. Information on demographics, clinical features, injuries, interventions and outcomes was noted. RESULTS: A total of 138 articles were identified from 1952 to 2019. There were 1072 children (males, 85.1%) and 1255 thoracoabdominal injuries. Mean age was 9.7 ± 3.3 years old. Common clinical features included abdominal pain and guarding, vomiting, fever and a handlebar imprint. The liver was the most frequently injured organ. Surgery was performed in 338 children with a mean age of 10.0 ± 3.3 years. Twenty-seven children (2.5%) were discharged and returned due to worsening symptoms, of whom 23 (85.2%) required surgery. Thirty-one children (2.9%) transferred to a higher level of care due to injury severity. Two deaths were reported. CONCLUSION: Bicycle handlebars can cause significant thoracoabdominal injuries. Presence of abdominal pain, vomiting, fever or a circular imprint on the chest or abdomen should prompt further workup. Future studies on diagnostic modalities and best practices are needed to lower the chance of missed injuries

Design and Development of Micro Off-grid Inverter for Solar Photovoltaic System using Proteus Simulation

530-539Solar photovoltaic panel generates DC power. It needs to be converted to AC power since most electrical appliances used in our day to day life runs on AC power supply. The conversion of DC to AC power happens using an inverter comprising of multiple electronic components, which generates harmonics in the AC power systems. The present investigation is focused to design a micro off-grid solar inverter with a minimal number of components using Proteus design suite simulation to generate quality power at an optimum cost. The designed solar inverter circuit mainly consists of resistors, capacitors, voltage regulators, CD 4047 IC, OP07 IC, R-C Circuit, MOSFETs and step-up transformer. The output of CD 4047 IC was given to the R-C three stage circuits for conversion from square to sinusoidal wave through amplifiers OP07 to boost the waveform voltage from 3.5 V to 7.0 V. The amplified sinusoidal waveform signal was applied to the gate of MOSFET combinations. Each set consists of two MOSFETs connected in parallel to produce voltage with 180° out of phase waveforms at MOSFET drain terminal and then supplied to center tapped step-up-transformer to produce constant 230 V AC output voltage at 50 Hz. The voltage regulation was achieved with the help of a DC-DC Boost Converter which makes the system capable of giving reliable power at 230 V even for varying solar irradiation from 145 W/m2 and above. Frequency regulation was achieved by varying the values of R and C across pins 1, 2 and 3 of CD 4047 IC. The maximum efficiency of the developed micro off-grid solar inverter’s hardware circuit was found to be 93.49% based on experimental measurements and 95.72% based on the simulation studies

Sizing Optimization and Techno-Economic Analysis of a Hybrid Renewable Energy System Using HOMER Pro Simulation

777-784Hybrid renewable energy system (HRES) is the integration of multiple energy generating systems installed to generate energy from the renewable sources such as Solar Photovoltaic (PV), wind, bio-generator etc. It is very prudent to determine viable combinations, optimum sizing and to have a techno-economic analysis of HRES before its procurement and installation. In this study two optimum system was modelled using HOMER Pro (open-source version) to meet the electrical load demand of an institution located in Chennai (12.59°N and 80.14°E) and to get the minimum Net Present Cost (NPC) of the proposed system. Based on the modeling, for optimum-1 system the capacity factor and contribution percentage of PV, wind turbine and bio-generator was found to be 79.41%, 0.98%, 19.61% and 92.83%, 0.43%, 6.75%, respectively. And for optimum-2 system, it was 85.86%, 2.02%, 12.12% and 93.31%, 0.82%, 5.87%, respectively. Sensitivity analysis was carried out to find out the variations on NPC and Cost of Energy (COE) by adjusting the cost of PV, wind turbine, battery, converter, bio-generator and fuel, from 0.8 to 1.2 times of its present cost. Based on HOMER Pro simulation, the most feasible system obtained was optimum-1 which consists of 81 kW photovoltaics, 1 kW wind turbine, 20 kW biogas generator, 47.3 kW converter and lead acid battery (rated 101 Ah/12 V-150 numbers) with NPC at $ 1,84,687 to generate energy 1,35,978 kWh per year