On the Injective Equitable Domination of Graphs

A dominating set D in a graph G is called an injective equitable dominating set (Inj-equitable dominating set) if for every , there exists such that u is adjacent to v and . The minimum cardinality of such a dominating set is denoted by and is called the Inj-equitable domination number of G. In this paper, we introduce the injective equitable domination of a graph and study its relation with other domination parameters. The minimal injective equitable dominating set, the injective equitable independence number , and the injective equitable domatic number are defined.

ON EQUITABLE VERTEX DISTINGUISHING EDGE COLORINGS OF TREES

It has been known that determining the exact value of vertex distinguishing edge index X ＇8（G） of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete, graphs, and graphs with maximum degree 2. Let rid（G） denote the number of vertices of degree d in G, and let X＇es（G） be the equitable vertex distinguishing edge index of G. We show that a tree T holds nl （T） ≤ X ＇s （T） ≤ n1 （T） ＋ 1 and X＇s（T） = X＇es（T） if T satisfies one of the following conditions （i） n2（T） ≤△（T） or （ii） there exists a constant c with respect to 0 〈 c 〈 1 such that n2（T） △ cn1（T） and ∑3 ≤d≤△（T）nd（T） ≤ （1 - c）n1（T） ＋ 1.

A Study on the Inj-Equitable Graph of a Graph

For any graph G, the Inj-equitable graph of a graph G, denoted by IE (G) , is the graph with the same vertices as G and for any two adjacent vertices u and v in IE (G), ≤ 1, where for any vertex w∈V (G) , degin (w)?= . In this paper, Inj-equitable graphs of some graphs are obtained, and some properties and results are established. Moreover, complete Inj-equitable graph and the Inj-equitable graph are defined.

The Middle Equitable Dominating Graphs

Let G= (V, E) be a graph and A(G) is the collection of all minimal equitable dominating set of G. The middle equitable dominating graph of G is the graph denoted by Med(G) with vertex set the disjoint union of V∪A(G) and (u, v) is an edge if and only if u ∩ v ≠ φ whenever u, v ∈ A(G) or u ∈ v whenever u ∈ v and v ∈ A(G) . In this paper, characterizations are given for graphs whose middle equitable dominating graph is connected and Kp∈Med(G) . Other properties of middle equitable dominating graphs are also obtained.

Some Results on Vertex Equitable Labeling

Let G be a graph with p vertices and q edges and let A= vertex labeling is said to be a vertex equitable labeling of G if it induces an edge labeling given by such that and , where is the number of vertices v with for A graph G is said to be a vertex equitable graph if it admits vertex equitable labeling. In this paper, we establish the vertex equitable labeling of a Tp-tree, where T is a Tp-tree with even number of vertices, bistar the caterpillar and

Universal Health Coverage in Somalia: Charting the Path to Equitable Healthcare Financing and Governance

Somalia is a country facing numerous challenges in achieving universal health coverage (UHC) and ensuring adequate healthcare financing, This article explores the complexities and obstacles that Somalia must overcome in its pursuit of UHC, the paper begins by providing an overview of the current healthcare landscape in Somalia, highlighting the lack of infrastructure, political instability, and limited financial resources that hinder the establishment of a comprehensive and equitable healthcare system. It then examines the role of international aid and non-governmental organizations (NGOs) in filling the healthcare gap, while emphasizing the need for a more sustainable, domestically financed solution. Drawing on a range of data sources and case studies, the article proposes a multi-faceted approach to strengthen healthcare governance, improve resource allocation, and foster local capacity building, the study delves into the unique obstacles that Somalia faces, including a lack of infrastructure, political instability, and limited financial resources, which hinder the establishment of a comprehensive and equitable healthcare system. The paper also examines the role of international aid and non-governmental organizations (NGOs) in filling the healthcare gap, while highlighting the need for a more sustainable, domestically financed solution. The findings underscore the importance of political commitment, international cooperation, and innovative financing mechanisms in advancing towards UHC in Somalia, providing valuable insights for other low resource, conflict affected settings.

The Equitable Total Chromatic Number of Some Join graphs

A proper total-coloring of graph G is said to be?equitable if the number of elements (vertices and edges) in any?two color classes differ by at most one, which the required?minimum number of colors is called the equitable total chromatic?number. In this paper, we prove some theorems on equitable?total coloring and derive the equitable total chromatic numbers?of Pm V?Sn, Pm V?Fn and Pm V Wn.

Regulation of Private Security Companies and Equitable Policing in Nigeria

Private security companies through the services they provide complement the police in crime prevention, order maintenance, and general improvement of security in societies thereby contributing to sustainable national development In order to fully benefit from their contribution and prevent them from engaging in illegal activities, states formulate and enforce policies that seek to regulate their activities. This paper examines the regulation of private guards companies in Nigeria using Abuja as its empirical core. The main aim is to promote understanding of how the private security industry is regulated and evaluate the effectiveness of the existing regulatory policy. Data for the paper were generated from in-depth interviews with the NSCDC （Nigeria Security and Civil Defence Corps）, analysis of the policy for PSCs （private security companies）, and a review of the annual performance report of the regulatory agency. The results revealed that the current system of PSC regulation is characterized by a lack of specialty classification of private security licences, limited regulatory scope with a focus on licensing of companies, lack of uniform standards on training, and high cost/difficulties in obtaining operational license. The paper concludes that the current regulatory setup for PSCs in Nigeria is ineffective and unsustainable and recommends an urgent review of the legal framework to alien its provisions with global standards and practices in the private security industry.

Does the China Railway Express Promote Equitable Regional Development?Evidence from Manufacturing Relocation

The China Railway Express(CRE)has improved regional economic integration with external markets and has reshaped regional development by encouraging the relocation of manufacturing activities("manufacturing transfer").We analyzed data from 284 prefecture-level cities in China to evaluate the impact of the CRE on regional manufacturing relocation,using a staggered difference-in-diferences model with entropy balancing.We found that the launch of the CRE significantly stimulated the relocation of manufacturing industries to node cities-key hubs for transportation and logistics within the CRE network-especially in the central,western,and northeastern regions.Specifically,the CRE significantly increased manufacturing dynamism,firm profitability,openness,and foreign investment in node cities,which attracted more manufacturing industries to relocate in this area.Higher levels of infrastructure development moderated the CRE's effect.Further analysis showed that significant moderating effects of infrastructure development occurred only in the western region.Overall,our conclusions provide useful insights into how to promote equitable regional development through manufacturing development in China.

Quest for Equitable Education in Phases:Insights From an NGO in China

Highlights Among the East Asian nations,a recurring predicament faced by educational institutions is that of providing inclusive but high-quality education.Active involvement of non-governmental organizations(NGOs)in education is valuable in China.Adream was such an NGO on education in China,established in 2008 with a singular and noble objective:promotion of equitable access to quality education within the disadvantaged regions of China.

Equitable Total Coloring of Some Join Graphs

The total chromatic number χt(G) of a graph G(V,E) is the minimum number of total independent partition sets of V E, satisfying that any two sets have no common element. If the difference of the numbers of any two total independent partition sets of V E is no more than one, then the minimum number of total independent partition sets of V E is called the equitable total chromatic number of G, denoted by χet(G). In this paper, we have obtained the equitable total chromatic number of Wm Kn, Fm Kn and Sm Kn whi...

Equitable Total Coloring of F_n ∨ W_n

The minimum number of total independent partition sets of V ∪ E of graph G（V,E） is called the total chromatic number of G denoted by χt（G）. If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one＇, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χet（G）. In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order.

Vertex Distinguishing Equitable Total Chromatic Number of Join Graph

A vertex distinguishing equitable total coloring of graph G is a proper total coloring of graph G such that any two distinct vertices＇ coloring sets are not identical and the difference of the elements colored by any two colors is not more than 1. In this paper we shall give vertex distinguishing equitable total chromatic number of join graphs Pn VPn, Cn VCn and prove that they satisfy conjecture 3, namely, the chromatic numbers of vertex distinguishing total and vertex distinguishing equitable total are the same for join graphs Pn V Pn and Cn ∨ Cn.

EQUITABLE COLORINGS OF LINE GRAPHS AND COMPLETE r-PARTITE GRAPHS

It is shown in this paper that Meyer’s conjecture on the equitable coloring holds for line graphs and complete r-partite graphs.

Optimization of equitable irrigation water delivery for a large-scale rice irrigation scheme

Equitable water allocation is essential in an irrigation scheme for obtaining potential crop yields from the entire scheme,especially when water supply is inadequate.An optimization model achieved this goal by coupling an optimal water allocation model with available water supply and irrigation water demand for a river-fed rice irrigation system in Malaysia.This model consists of a paddy field water balance module and an optimization module.The outputs from the module are daily irrigation demand and surface runoff,if there is any.The optimization module consists of an objective function,which minimizes water shortage across the scheme area while maintaining equity in water allocation.This model performs optimization subject to several system constraints,and the decision variable of the model is daily releases or supply to the tertiary canals.Performance of this model remained unaffected under different water supply conditions,and the optimization model reliably examined the effects of alternate water allocation and management rules with field information.It improves efficiency and equity in water allocation with respect to crop growth stages and water shortages rather than simply cutting irrigation supply on a proportional basis to overcome water shortages.

On the Adjacent Vertex-distinguishing Equitable Edge Coloring of Graphs

Let G（V, E） be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if （1） C（u）≠C（v） for uv ∈ E（G）, where C（u） = {f（uv）｜uv ∈ E}, and （2） for any i, j = 1, 2,… k, we have ｜｜Ei｜ ｜Ej｜｜ ≤ 1, where Ei = {e｜e ∈ E（G） and f（e） = i}. χáve （G） = min{k｜ there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χ áve （G） of some special graphs and present a conjecture.

Equitable Coloring of Three Classes of 1-planar Graphs

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near-independent crossings or independent crossings, say NIC-planar graph or IC-planar graph, is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex or no common vertices, respectively. In this paper, we prove that each 1-planar graph, NIC-planar graph or IC-planar graph with maximum degree A at least 15, 13 or 12 has an equitable △-coloring, respectively. This verifies the well-known Chen-Lih-Wu Conjecture for three classes of 1-planar graphs and improves some known results.

Equitable Vertex Arboricity Conjecture Holds for Graphs with Low Degeneracy

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations.Namely,an equitable tree-Zc-coloring of a graph is a vertex coloring using k distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one.In this paper,we show some theoretical results on the equitable tree-coloring of graphs by proving that every d-degenerate graph with maximum degree at most Δ is equitably tree-fc-colorable for every integer k≥(Δ+1)/2 provided that Δ≥9.818d,confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.

Equitable Presentations for Multiparameter Quantum Groups

In this paper, we give an equitable presentation for the multiparameter quantum group associated to a symmetrizable Kac–Moody Lie algebra, which can be regarded as a natural generalization of the Terwilliger's equitable presentation for the one-parameter quantum group.

Piecewise w∞-equitable Efficiency in Multiobjective Programming

In equitable multiobjective optimization all the objectives are uniformly optimized, but in some cases the decision maker believes that some of them should be uniformly optimized according to the importance of objectives. To solve this problem in this paper, the original problem is decomposed into a collection of smaller subproblems, according to the decision maker, and the subproblems are solved by the concept of wr- equitable efficiency, where w ∈ R＋ m is a weight vector. First some theoretical and practical aspects of Pwr- equitably efficient solutions are discussed and by using the concept of Pwr-equitable efficiency one model is presented to coordinate weakly wr-equitable efficient solutions of subproblems. Then the concept of Pw ∞- equitable is introduced to generate subsets of equitably efficient solutions, which aims to offer a limited number of representative solutions to the decision maker.