An f-edge cover-colouring of a graph G = (V, E) is an assignment of colours to the edges of G such that every colour appears at each vertex v ∈ V at least f(v) times. The maximum number of colours needed to f-edge co...An f-edge cover-colouring of a graph G = (V, E) is an assignment of colours to the edges of G such that every colour appears at each vertex v ∈ V at least f(v) times. The maximum number of colours needed to f-edge cover colour G is called the f-edge cover chromatic index of G, denoted by Xfc(G). This paper gives that min[d(v)-1/f(v)]≤xfc(G)≤展开更多
Graph colouring is the system of assigning a colour to each vertex of a graph.It is done in such a way that adjacent vertices do not have equal colour.It is fundamental in graph theory.It is often used to solve real-w...Graph colouring is the system of assigning a colour to each vertex of a graph.It is done in such a way that adjacent vertices do not have equal colour.It is fundamental in graph theory.It is often used to solve real-world problems like traffic light signalling,map colouring,scheduling,etc.Nowadays,social networks are prevalent systems in our life.Here,the users are considered as vertices,and their connections/interactions are taken as edges.Some users follow other popular users’profiles in these networks,and some don’t,but those non-followers are connected directly to the popular profiles.That means,along with traditional relationship(information flowing),there is another relation among them.It depends on the domination of the relationship between the nodes.This type of situation can be modelled as a directed fuzzy graph.In the colouring of fuzzy graph theory,edge membership plays a vital role.Edge membership is a representation of flowing information between end nodes of the edge.Apart from the communication relationship,there may be some other factors like domination in relation.This influence of power is captured here.In this article,the colouring of directed fuzzy graphs is defined based on the influence of relationship.Along with this,the chromatic number and strong chromatic number are provided,and related properties are investigated.An application regarding COVID-19 infection is presented using the colouring of directed fuzzy graphs.展开更多
A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in ...A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in an acyclic edge colouring of G. In this paper, we discuss the acyclic edge colouring of middle, central, total and line graphs of prime related star graph families. Also exact values of acyclic chromatic indices of such graphs are derived and some of their structural properties are discussed.展开更多
Colouring of the austenitic stainless steel alloy （20.45% Cr, 8.57% Ni） was carried out in NaNO3-KNO3 eutectic melt without and with additions of Na2O2, NaCl and their mixtures at different temperatures ranging from...Colouring of the austenitic stainless steel alloy （20.45% Cr, 8.57% Ni） was carried out in NaNO3-KNO3 eutectic melt without and with additions of Na2O2, NaCl and their mixtures at different temperatures ranging from 400-600℃, under open-circuit and galvanostatic anodic polarization conditions. The produced colours greatly depend on the thickness of oxide films, which in turn depends on the composition of the molten bath and its temperature. The more attractive, bright, adherent and uniform coloured oxide films can be obtained at 400, 450 and 500℃ in molten nitrate bath containing NaCl and Na2O2 mixtures. The pitting corrosion susceptibility of the coloured oxide films was tested in FeCl3 and NaCl as corrosive media. The obtained results indicate that the pitting corrosion susceptibility of the coloured oxide films greatly depends on the previous operating conditions of the colouring process of the stainless steel specimens such as the composition of molten bath, temperature and technique of colouring process.展开更多
Several possible definitions of local injectivity for a homomorphism of an oriented graph G to an oriented graph H are considered. In each case, we determine the complexity of deciding whether there exists such a homo...Several possible definitions of local injectivity for a homomorphism of an oriented graph G to an oriented graph H are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when G is given and H is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.展开更多
Many of the primary school students are interested in painting.However, few of them know how to learn painting.Many primary school students think that they can begin their painting by imitating.In fact, the first step...Many of the primary school students are interested in painting.However, few of them know how to learn painting.Many primary school students think that they can begin their painting by imitating.In fact, the first step to learn painting is to colour the pictures, for you can make the pictures more beautiful by colouring them.Also, you can feel how the pictures were drawn by colouring them.展开更多
A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We co...A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclie k-eolourings such that each colour class induces a graph with a given （hereditary） property. In particular, we consider aeyclic k-eolourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyelic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with A（G） ≤ 4 can be acyelically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.展开更多
Hot-dip galvanizing is a widely used technique to prevent steel from oxidizing in air.However, only silver-grey zinc coating can no longer meet the need of people. At theWorld’s Fair in July 1989, Japanese first exhi...Hot-dip galvanizing is a widely used technique to prevent steel from oxidizing in air.However, only silver-grey zinc coating can no longer meet the need of people. At theWorld’s Fair in July 1989, Japanese first exhibited blue, red, yellow green andbrown hot-dip zinc coating sheets. These patent coatings are more incorruptible thantraditiona ones. This note tries to reveal the colouring mechanism of these coatings.展开更多
Flower constancy describes the phenomenon that pollinators tend to successively visit flowers of a single species during foraging,reducing reproductive interference in natural communities.The extent of flower constanc...Flower constancy describes the phenomenon that pollinators tend to successively visit flowers of a single species during foraging,reducing reproductive interference in natural communities.The extent of flower constancy is largely determined by the floral traits of co-flowering species.Both higher inter-specific and lower intraspecific differences of floral traits should contribute to a higher level of flower constancy.However,previous studies mainly focused on interspecific difference,and the intraspecific variation(consistency)of floral traits received much less attention.We hypothesise that selection may favour lower intraspecific floral trait variation in communities composed of multiple co-flowering congeners.We investigated the floral colour variation of three focal Pedicularis species that share pollinators in 19communities composed of either single or multiple Pedicularis species.Colour was quantified using image-based colour analysis as perceived by pollinators.We found that most of the intrapopulation floral colour variation was below the colour discrimination threshold of bumblebees,implying strongly constrained by the visual selection by pollinators.Contrary to the hypothesis,there is no significant difference in intraspecific floral colour variation between different community contexts.It may be due to the relatively large interspecific floral colour differences of most co-flowering species.The influence of community context on intraspecific variation may be reflected in floral traits other than colours.展开更多
ObjectiveThe study aimed to explore the factors regulating carotenoid accumulation in flesh color. MethodA loquat mutation （red-or orange-fleshed plant emerged a bud mutation of white-flesh in trunk） was used as mat...ObjectiveThe study aimed to explore the factors regulating carotenoid accumulation in flesh color. MethodA loquat mutation （red-or orange-fleshed plant emerged a bud mutation of white-flesh in trunk） was used as material; HPLC analysis of β-carotene content was conducted. ResultThe β-carotene concentration in the flesh of wild and mutant types was 60.9 and 4.6 μg/g fresh weight, respectively. According to the conserved regions of genes from rose family genome, carotenogenic gene fragments in wild and mutant types were obtained. No nucleotide variation of the carotenogenic gene fragments was observed between wild and mutant genome. Real-time quantitative polymerase chain reaction （Q-PCR） was compared and one carotenogenic gene, β-ring hydroxylase （HYB） were considerably suppressed in mature mutant loquat fruits compared with that in wild. The other six carotenogenic genes were also expressed but the expression patterns appeared to be not correlated with the amount of β-carotene concentration in wild loquat flesh. ConclusionThe mutant whitish loquat lacks the ability to synthesize β-carotene because of the transcriptional down-regulation of carotenogenic gene HYB.展开更多
The concept of graphlike manifolds was presented in [1] and the problem of counting the homeomorphic classes of graphlike manifolds has been studied in a series of articles, e.g., [1～8]. In this paper we suggest an a...The concept of graphlike manifolds was presented in [1] and the problem of counting the homeomorphic classes of graphlike manifolds has been studied in a series of articles, e.g., [1～8]. In this paper we suggest an approach based on the graph colouring, Abelian group and the combinatorial enumeration method.展开更多
This paper presents a high-order coupled compact integrated RBF(CC IRBF)approximation based domain decomposition(DD)algorithm for the discretisation of second-order differential problems.Several Schwarz DD algorithms,...This paper presents a high-order coupled compact integrated RBF(CC IRBF)approximation based domain decomposition(DD)algorithm for the discretisation of second-order differential problems.Several Schwarz DD algorithms,including one-level additive/multiplicative and two-level additive/multiplicative/hybrid,are employed.The CCIRBF based DD algorithms are analysed with different mesh sizes,numbers of subdomains and overlap sizes for Poisson problems.Our convergence analysis shows that the CCIRBF two-level multiplicative version is the most effective algorithm among various schemes employed here.Especially,the present CCIRBF two-level method converges quite rapidly even when the domain is divided into many subdomains,which shows great promise for either serial or parallel computing.For practical tests,we then incorporate the CCIRBF into serial and parallel two-level multiplicative Schwarz.Several numerical examples,including those governed by Poisson and Navier-Stokes equations are analysed to demonstrate the accuracy and efficiency of the serial and parallel algorithms implemented with the CCIRBF.Numerical results show:(i)the CCIRBF-Serial and-Parallel algorithms have the capability to reach almost the same solution accuracy level of the CCIRBF-Single domain,which is ideal in terms of computational calculations;(ii)the CCIRBF-Serial and-Parallel algorithms are highly accurate in comparison with standard finite difference,compact finite difference and some other schemes;(iii)the proposed CCIRBF-Serial and-Parallel algorithms may be used as alternatives to solve large-size problems which the CCIRBF-Single domain may not be able to deal with.The ability of producing stable and highly accurate results of the proposed serial and parallel schemes is believed to be the contribution of the coarse mesh of the two-level domain decomposition and the CCIRBF approximation.It is noted that the focus of this paper is on the derivation of highly accurate serial and parallel algorithms for second-order differential problems.The scope of this work does not cover a thorough analysis of computational time.展开更多
Eucommia ulmoides‘Hongye’is a new ornamental variety of E.ulmoides with excellent red or purple foliage.We found that E.ulmoides‘Hongye’exhibited a gradual change from green to red colour under light conditions.Ho...Eucommia ulmoides‘Hongye’is a new ornamental variety of E.ulmoides with excellent red or purple foliage.We found that E.ulmoides‘Hongye’exhibited a gradual change from green to red colour under light conditions.However,the colouring mechanism in the leaves of E.ulmoides‘Hongye’remains unclear.In this study,we compared the pigment content and leaf colour index of E.ulmoides‘Hongye’at five stages with those of E.ulmoides‘Xiaoye’,which was used as the control variety.The transcriptome sequencing data of the first-period(H1,green)and fifth-period(H5,red)leaves were also analysed and compared.The corresponding gene regulation in anthocyanin-related metabolic pathways was then analysed.Physiological results indicated that the contents of flavonoids and anthocyanins in red leaves(H5)were significantly higher than those in green leaves(H1),whereas the chlorophyll content in red leaves(H5)was lower than that in green leaves(H1).Moreover,the carotenoid content did not significantly differ between the two varieties.A transcriptome analysis identified 4240 differentially expressed genes(DEGs),and 20 of these genes were found to be involved in flavonoid and anthocyanin biosynthesis pathways.The results provide a reference for further study of the leaf colouration mechanism in E.ulmoides.展开更多
By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms...By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.展开更多
文摘An f-edge cover-colouring of a graph G = (V, E) is an assignment of colours to the edges of G such that every colour appears at each vertex v ∈ V at least f(v) times. The maximum number of colours needed to f-edge cover colour G is called the f-edge cover chromatic index of G, denoted by Xfc(G). This paper gives that min[d(v)-1/f(v)]≤xfc(G)≤
基金supported and funded by the Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(2018R1D1A1B07049321).
文摘Graph colouring is the system of assigning a colour to each vertex of a graph.It is done in such a way that adjacent vertices do not have equal colour.It is fundamental in graph theory.It is often used to solve real-world problems like traffic light signalling,map colouring,scheduling,etc.Nowadays,social networks are prevalent systems in our life.Here,the users are considered as vertices,and their connections/interactions are taken as edges.Some users follow other popular users’profiles in these networks,and some don’t,but those non-followers are connected directly to the popular profiles.That means,along with traditional relationship(information flowing),there is another relation among them.It depends on the domination of the relationship between the nodes.This type of situation can be modelled as a directed fuzzy graph.In the colouring of fuzzy graph theory,edge membership plays a vital role.Edge membership is a representation of flowing information between end nodes of the edge.Apart from the communication relationship,there may be some other factors like domination in relation.This influence of power is captured here.In this article,the colouring of directed fuzzy graphs is defined based on the influence of relationship.Along with this,the chromatic number and strong chromatic number are provided,and related properties are investigated.An application regarding COVID-19 infection is presented using the colouring of directed fuzzy graphs.
文摘A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in an acyclic edge colouring of G. In this paper, we discuss the acyclic edge colouring of middle, central, total and line graphs of prime related star graph families. Also exact values of acyclic chromatic indices of such graphs are derived and some of their structural properties are discussed.
文摘Colouring of the austenitic stainless steel alloy （20.45% Cr, 8.57% Ni） was carried out in NaNO3-KNO3 eutectic melt without and with additions of Na2O2, NaCl and their mixtures at different temperatures ranging from 400-600℃, under open-circuit and galvanostatic anodic polarization conditions. The produced colours greatly depend on the thickness of oxide films, which in turn depends on the composition of the molten bath and its temperature. The more attractive, bright, adherent and uniform coloured oxide films can be obtained at 400, 450 and 500℃ in molten nitrate bath containing NaCl and Na2O2 mixtures. The pitting corrosion susceptibility of the coloured oxide films was tested in FeCl3 and NaCl as corrosive media. The obtained results indicate that the pitting corrosion susceptibility of the coloured oxide films greatly depends on the previous operating conditions of the colouring process of the stainless steel specimens such as the composition of molten bath, temperature and technique of colouring process.
文摘Several possible definitions of local injectivity for a homomorphism of an oriented graph G to an oriented graph H are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when G is given and H is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.
文摘Many of the primary school students are interested in painting.However, few of them know how to learn painting.Many primary school students think that they can begin their painting by imitating.In fact, the first step to learn painting is to colour the pictures, for you can make the pictures more beautiful by colouring them.Also, you can feel how the pictures were drawn by colouring them.
基金supported by the Minister of Science and Higher Education of Poland(Grant No.JP2010009070)
文摘A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclie k-eolourings such that each colour class induces a graph with a given （hereditary） property. In particular, we consider aeyclic k-eolourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyelic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with A（G） ≤ 4 can be acyelically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.
文摘Hot-dip galvanizing is a widely used technique to prevent steel from oxidizing in air.However, only silver-grey zinc coating can no longer meet the need of people. At theWorld’s Fair in July 1989, Japanese first exhibited blue, red, yellow green andbrown hot-dip zinc coating sheets. These patent coatings are more incorruptible thantraditiona ones. This note tries to reveal the colouring mechanism of these coatings.
基金funded by the West Light Foundation of The Chinese Academy of Sciencesnational youth talent support programYunnan youth talents plan(YNWR-QNBJ-2018-183 to Y.N.)。
文摘Flower constancy describes the phenomenon that pollinators tend to successively visit flowers of a single species during foraging,reducing reproductive interference in natural communities.The extent of flower constancy is largely determined by the floral traits of co-flowering species.Both higher inter-specific and lower intraspecific differences of floral traits should contribute to a higher level of flower constancy.However,previous studies mainly focused on interspecific difference,and the intraspecific variation(consistency)of floral traits received much less attention.We hypothesise that selection may favour lower intraspecific floral trait variation in communities composed of multiple co-flowering congeners.We investigated the floral colour variation of three focal Pedicularis species that share pollinators in 19communities composed of either single or multiple Pedicularis species.Colour was quantified using image-based colour analysis as perceived by pollinators.We found that most of the intrapopulation floral colour variation was below the colour discrimination threshold of bumblebees,implying strongly constrained by the visual selection by pollinators.Contrary to the hypothesis,there is no significant difference in intraspecific floral colour variation between different community contexts.It may be due to the relatively large interspecific floral colour differences of most co-flowering species.The influence of community context on intraspecific variation may be reflected in floral traits other than colours.
基金Supported by Special Fund for Agro-scientific Research in the Public Interest of China(201003073)Key Laboratory Program of Agriculture Ministry of China(2013JCYJ-004)+1 种基金Applied Basic Research Program of Chengdu City Science and Technology Bureau(11DXYB039NC)Youth Foundation of Sichuan Province(2011QNJJ-010)~~
文摘ObjectiveThe study aimed to explore the factors regulating carotenoid accumulation in flesh color. MethodA loquat mutation （red-or orange-fleshed plant emerged a bud mutation of white-flesh in trunk） was used as material; HPLC analysis of β-carotene content was conducted. ResultThe β-carotene concentration in the flesh of wild and mutant types was 60.9 and 4.6 μg/g fresh weight, respectively. According to the conserved regions of genes from rose family genome, carotenogenic gene fragments in wild and mutant types were obtained. No nucleotide variation of the carotenogenic gene fragments was observed between wild and mutant genome. Real-time quantitative polymerase chain reaction （Q-PCR） was compared and one carotenogenic gene, β-ring hydroxylase （HYB） were considerably suppressed in mature mutant loquat fruits compared with that in wild. The other six carotenogenic genes were also expressed but the expression patterns appeared to be not correlated with the amount of β-carotene concentration in wild loquat flesh. ConclusionThe mutant whitish loquat lacks the ability to synthesize β-carotene because of the transcriptional down-regulation of carotenogenic gene HYB.
文摘The concept of graphlike manifolds was presented in [1] and the problem of counting the homeomorphic classes of graphlike manifolds has been studied in a series of articles, e.g., [1～8]. In this paper we suggest an approach based on the graph colouring, Abelian group and the combinatorial enumeration method.
文摘This paper presents a high-order coupled compact integrated RBF(CC IRBF)approximation based domain decomposition(DD)algorithm for the discretisation of second-order differential problems.Several Schwarz DD algorithms,including one-level additive/multiplicative and two-level additive/multiplicative/hybrid,are employed.The CCIRBF based DD algorithms are analysed with different mesh sizes,numbers of subdomains and overlap sizes for Poisson problems.Our convergence analysis shows that the CCIRBF two-level multiplicative version is the most effective algorithm among various schemes employed here.Especially,the present CCIRBF two-level method converges quite rapidly even when the domain is divided into many subdomains,which shows great promise for either serial or parallel computing.For practical tests,we then incorporate the CCIRBF into serial and parallel two-level multiplicative Schwarz.Several numerical examples,including those governed by Poisson and Navier-Stokes equations are analysed to demonstrate the accuracy and efficiency of the serial and parallel algorithms implemented with the CCIRBF.Numerical results show:(i)the CCIRBF-Serial and-Parallel algorithms have the capability to reach almost the same solution accuracy level of the CCIRBF-Single domain,which is ideal in terms of computational calculations;(ii)the CCIRBF-Serial and-Parallel algorithms are highly accurate in comparison with standard finite difference,compact finite difference and some other schemes;(iii)the proposed CCIRBF-Serial and-Parallel algorithms may be used as alternatives to solve large-size problems which the CCIRBF-Single domain may not be able to deal with.The ability of producing stable and highly accurate results of the proposed serial and parallel schemes is believed to be the contribution of the coarse mesh of the two-level domain decomposition and the CCIRBF approximation.It is noted that the focus of this paper is on the derivation of highly accurate serial and parallel algorithms for second-order differential problems.The scope of this work does not cover a thorough analysis of computational time.
基金Natural Science Foundation of Henan Province of China(202300410554)Key R&D and Promotion Project of Henan Province(Science and Technology Research)(192102110169,202102110229)].
文摘Eucommia ulmoides‘Hongye’is a new ornamental variety of E.ulmoides with excellent red or purple foliage.We found that E.ulmoides‘Hongye’exhibited a gradual change from green to red colour under light conditions.However,the colouring mechanism in the leaves of E.ulmoides‘Hongye’remains unclear.In this study,we compared the pigment content and leaf colour index of E.ulmoides‘Hongye’at five stages with those of E.ulmoides‘Xiaoye’,which was used as the control variety.The transcriptome sequencing data of the first-period(H1,green)and fifth-period(H5,red)leaves were also analysed and compared.The corresponding gene regulation in anthocyanin-related metabolic pathways was then analysed.Physiological results indicated that the contents of flavonoids and anthocyanins in red leaves(H5)were significantly higher than those in green leaves(H1),whereas the chlorophyll content in red leaves(H5)was lower than that in green leaves(H1).Moreover,the carotenoid content did not significantly differ between the two varieties.A transcriptome analysis identified 4240 differentially expressed genes(DEGs),and 20 of these genes were found to be involved in flavonoid and anthocyanin biosynthesis pathways.The results provide a reference for further study of the leaf colouration mechanism in E.ulmoides.
文摘By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.
