Eighty years ago,the international community,reflecting deeply on the painful lessons of the two devastating world wars,established the United Nations,ushering in a new chapter of global governance and making historic...Eighty years ago,the international community,reflecting deeply on the painful lessons of the two devastating world wars,established the United Nations,ushering in a new chapter of global governance and making historic contributions to maintaining world peace and development.展开更多
Cameroon’s capital Yaoundécame alive on 16 and 17 July as it hosted a major international gathering for the liquefied natural gas(LNG)sector.The event brought together over 1,000 industry professionals and 50 ex...Cameroon’s capital Yaoundécame alive on 16 and 17 July as it hosted a major international gathering for the liquefied natural gas(LNG)sector.The event brought together over 1,000 industry professionals and 50 exhibitors from more than 30 countries including China to address the future of cleaner and more inclusive energy.展开更多
The equitable total coloring of a graph G is a total coloring such that the numbers of elements in any two colors differ by at most one.The smallest number of colors needed for an equitable total coloring is called th...The equitable total coloring of a graph G is a total coloring such that the numbers of elements in any two colors differ by at most one.The smallest number of colors needed for an equitable total coloring is called the equitable total chromatic number.This paper contributes to the equitable total coloring of Fibonacci graphs F_(∆,n).We determine the equitable total chromatic numbers of F_(∆,n) for∆=3,4,5 and propose a conjecture on that for∆>=6.展开更多
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 dominatin...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.展开更多
For a proper edge coloring c of a graph G, if the sets of colors of adjacent vertices are distinct, the edge coloring c is called an adjacent strong edge coloring of G. Let ci be the number of edges colored by i. If [...For a proper edge coloring c of a graph G, if the sets of colors of adjacent vertices are distinct, the edge coloring c is called an adjacent strong edge coloring of G. Let ci be the number of edges colored by i. If [ci - cj] ≤1 for any two colors i and j, then c is an equitable edge coloring of G. The coloring c is an equitable adjacent strong edge coloring of G if it is both adjacent strong edge coloring and equitable edge coloring. The least number of colors of such a coloring c is called the equitable adjacent strong chromatic index of G. In this paper, we determine the equitable adjacent strong chromatic index of the joins of paths and cycles. Precisely, we show that the equitable adjacent strong chromatic index of the joins of paths and cycles is equal to the maximum degree plus one or two.展开更多
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...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.展开更多
An equitable(O^(1)_(k),O^(2)_(k),...,O^(m)_(k))-partition of a graph G,which is also called a k cluster m-partition,is the partition of V(G)into m non-empty subsets V_(1),V_(2),...,Vm such that for every integer i in{...An equitable(O^(1)_(k),O^(2)_(k),...,O^(m)_(k))-partition of a graph G,which is also called a k cluster m-partition,is the partition of V(G)into m non-empty subsets V_(1),V_(2),...,Vm such that for every integer i in{1,2,...,m},G[Vi]is a graph with components of order at most k,and for each distinct pair i,j in{1,...,m},there is−1≤|Vi|−|Vj|≤1.In this paper,we proved that every planar graph G with minimum degreeδ(G)≥2 and girth g(G)≥12 admits an equitable(O_(1)^(7),O^(2)_(7),...,O^(m)_(7))-partition,for any integer m≥2.展开更多
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)?...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.展开更多
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)...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.展开更多
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...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展开更多
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 ...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.展开更多
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 equit...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.展开更多
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 ...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.展开更多
In medical practice,the effectiveness of interventions often hinges on precise dosage control and strict scheduling,with adherence to prescribed doses being fundamental for patients to achieve optimal therapeutic outc...In medical practice,the effectiveness of interventions often hinges on precise dosage control and strict scheduling,with adherence to prescribed doses being fundamental for patients to achieve optimal therapeutic outcomes.1 However,globally,inaccurate or missing medical records have become a common concern.展开更多
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 to...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...展开更多
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...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.展开更多
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 couplin...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.展开更多
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...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.展开更多
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}, a...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.展开更多
文摘Eighty years ago,the international community,reflecting deeply on the painful lessons of the two devastating world wars,established the United Nations,ushering in a new chapter of global governance and making historic contributions to maintaining world peace and development.
文摘Cameroon’s capital Yaoundécame alive on 16 and 17 July as it hosted a major international gathering for the liquefied natural gas(LNG)sector.The event brought together over 1,000 industry professionals and 50 exhibitors from more than 30 countries including China to address the future of cleaner and more inclusive energy.
基金Supported by the National Natural Science Foundation of China(Grant No.62072292)the Natural Science Foundation of Shandong Province(Grant No.ZR2020KF010).
文摘The equitable total coloring of a graph G is a total coloring such that the numbers of elements in any two colors differ by at most one.The smallest number of colors needed for an equitable total coloring is called the equitable total chromatic number.This paper contributes to the equitable total coloring of Fibonacci graphs F_(∆,n).We determine the equitable total chromatic numbers of F_(∆,n) for∆=3,4,5 and propose a conjecture on that for∆>=6.
文摘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.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant Nos. 2011B019)the National Natural Science Foundation of China (Grant Nos. 10971144+2 种基金1110102011171026)the Natural Science Foundation of Beijing (Grant No. 1102015)
文摘For a proper edge coloring c of a graph G, if the sets of colors of adjacent vertices are distinct, the edge coloring c is called an adjacent strong edge coloring of G. Let ci be the number of edges colored by i. If [ci - cj] ≤1 for any two colors i and j, then c is an equitable edge coloring of G. The coloring c is an equitable adjacent strong edge coloring of G if it is both adjacent strong edge coloring and equitable edge coloring. The least number of colors of such a coloring c is called the equitable adjacent strong chromatic index of G. In this paper, we determine the equitable adjacent strong chromatic index of the joins of paths and cycles. Precisely, we show that the equitable adjacent strong chromatic index of the joins of paths and cycles is equal to the maximum degree plus one or two.
基金supported by the National Natural Science Foundation of China (61163054),supported by the National Natural Science Foundation of China (61163037)
文摘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.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1207126512271331)the Natural Science Foundation of Shandong Province(Grant No.ZR202102250232).
文摘An equitable(O^(1)_(k),O^(2)_(k),...,O^(m)_(k))-partition of a graph G,which is also called a k cluster m-partition,is the partition of V(G)into m non-empty subsets V_(1),V_(2),...,Vm such that for every integer i in{1,2,...,m},G[Vi]is a graph with components of order at most k,and for each distinct pair i,j in{1,...,m},there is−1≤|Vi|−|Vj|≤1.In this paper,we proved that every planar graph G with minimum degreeδ(G)≥2 and girth g(G)≥12 admits an equitable(O_(1)^(7),O^(2)_(7),...,O^(m)_(7))-partition,for any integer m≥2.
文摘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.
文摘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.
文摘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
文摘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.
文摘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.
文摘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.
基金supported by the National Key R&D Program of China(2022YFA1104600)the National Natural Science Foundation of China(82372377)+2 种基金the Biomaterials and Regenerative Medicine Institute Cooperative Research Project by Shanghai Jiao Tong University School of Medicine(grant number 2022LHBO8)the Shanghai Key Laboratory of Orthopaedic Implants,Department of Orthopaedics(no.KFKT202206)the Key R&D Program of Jiangsu Province Social Development Project(BE2022708).
文摘In medical practice,the effectiveness of interventions often hinges on precise dosage control and strict scheduling,with adherence to prescribed doses being fundamental for patients to achieve optimal therapeutic outcomes.1 However,globally,inaccurate or missing medical records have become a common concern.
基金the National Natural Science Foundation of China (No.10771091)
文摘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...
基金Supported by the National Natural Science Foundation of China(No.10771091)
文摘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.
文摘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.
基金supported by the Natural Science Basic Research Plan in Shaanxi Province of China(No.2017JM1010)the Fundamental Research Funds for the Central Universities(No.JB170706)+5 种基金the Specialized Research Fund for the Doctoral Program of Higher Education(No.20130203120021)the National Natural Science Foundation of China(No.11301410)the National Natural Science Foundation of China(No.11501316)the Shandong Provincial Natural Science Foundation,China(No.ZR2014AQ001)the China Postdoctoral Science Foundation(No.2015M570569)supported by the Natural Science Foundation of Xinjiang Province of China(No.2015211A003)
文摘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.
基金Supported by the National Natural Science Foundation of China(No.10771091No.61163010)Ningxia University Science Research Foundation(No.(E)ndzr09-15)
文摘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.
