A t-tone coloring of a graph assigns t distinct colors to each vertex with vertices at distance d having fewer than d colors in common.The t-tone chromatic number of a graph is the smallest number of colors used in al...A t-tone coloring of a graph assigns t distinct colors to each vertex with vertices at distance d having fewer than d colors in common.The t-tone chromatic number of a graph is the smallest number of colors used in all t-tone colorings of that graph.In this article,we study t-tone coloring of some finite planar lattices and obtain exact formulas for their t-tone chromatic number.展开更多
Image coloring is an inherently uncertain and multimodal problem.By inputting a grayscale image into a coloring network,visually plausible colored photos can be generated.Conventional methods primarily rely on semanti...Image coloring is an inherently uncertain and multimodal problem.By inputting a grayscale image into a coloring network,visually plausible colored photos can be generated.Conventional methods primarily rely on semantic information for image colorization.These methods still suffer from color contamination and semantic confusion.This is largely due to the limited capacity of convolutional neural networks to learn deep semantic information inherent in images effectively.In this paper,we propose a network structure that addresses these limitations by leveraging multi-level semantic information classification and fusion.Additionally,we introduce a global semantic fusion network to combat the issues of color contamination.The proposed coloring encoder accurately extracts object-level semantic information from images.To further enhance visual plausibility,we employ a self-supervised adversarial training method.We train the network structure on various datasets with varying amounts of data and evaluate its performance using the ImageNet validation set and COCO validation set.Experimental results demonstrate that our proposed algorithm can generate more realistic images compared to previous approaches,showcasing its high generalization ability.展开更多
A proper conflict-free k-coloring of a graph is a proper k-coloring in which each nonisolated vertex has a color that appears ex-actly once in its open neighborhood.A graph is PCF k-colorable if it admits a proper con...A proper conflict-free k-coloring of a graph is a proper k-coloring in which each nonisolated vertex has a color that appears ex-actly once in its open neighborhood.A graph is PCF k-colorable if it admits a proper conflict-free k-coloring.The PCF chromatic number of a graph G,denoted by χ_(pcf)(G),is the minimum k such that G is PCF k-colorable.Caro et al conjectured that for a connected graph G with maximum degreeΔ≥3,χ_(pcf)(G)≤Δ+1.One case in this conjecture,a connected graph with maximum degree 3 is PCF 4-colorable,can be derived from the result of Liu and Yu.Jiménez et al stated that the upper bound of PCF chromatic number of a graph G is max{5,x(G)}without a proof.In this paper,we give new proofs of the two results above and derive that for a connected graph G with maximum degreeΔ≥3,its complete subdivision is PCF(Δ+1)-colorable.展开更多
A proper edge t-coloring of a graph G is a coloring of its edges with colors 1, 2,..., t, such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of...A proper edge t-coloring of a graph G is a coloring of its edges with colors 1, 2,..., t, such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each vertex, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. In this paper, we provide a new proof of the result on the colors in cyclically interval edge colorings of simple cycles which was first proved by Rafayel R. Kamalian in the paper “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles, Open Journal of Discrete Mathematics, 2013, 43-48”.展开更多
Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints....Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C（x） denote the set of colors of vertex x and of the edges incident with x, we call C（x） the color set of x. If C（u） ≠ C（v） for any two different vertices u and v of V （G）, then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.展开更多
Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct verti...Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct vertices u and v of V（G）, then f is called a vertex- distinguishing total k-coloring of G. The minimum number k for which there exists a vertex- distinguishing total k-coloring of G is called the vertex-distinguishing total chromatic number of G and denoted by Xvt（G）. The vertex-disjoint union of two cycles of length n is denoted by 2Cn. We will obtain Xvt（2Cn） in this paper.展开更多
Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoi...Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.展开更多
Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges i...Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C（u） =fi C（v） for any two different vertices u and v of V（G）, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The ie iV., minimum number of colors required for a VDIET coloring of G is denoted by X,t[ 1, and it is called the VDIET chromatic number of G. We will give VDIET chromatic numbers for complete bipartite graph K4,n（n ≥ 4）, Kn,n （5 ≤ n ≤21） in this article.展开更多
Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges i...Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C（u）=C（v） for any two different vertices u and v of V （G）, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt（G）, and is called the VDIET chromatic number of G. We get the VDIET chromatic numbers of cycles and wheels, and propose related conjectures in this paper.展开更多
A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of ...A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv∈ E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by X＇Aa（G）, is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.展开更多
Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of verte...Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.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
We give the optimal I-(VI-)total colorings of mC_(4)which are vertex-distinguished by multiple sets by the use of the method of constructing a matrix whose entries are the suitable multiple sets or empty sets and the ...We give the optimal I-(VI-)total colorings of mC_(4)which are vertex-distinguished by multiple sets by the use of the method of constructing a matrix whose entries are the suitable multiple sets or empty sets and the method of distributing color set in advance.Thereby we obtain I-(VI-)total chromatic numbers of mC_(4)which are vertex-distinguished by multiple sets.展开更多
After a necessary condition is given, 3-rainbow coloring of split graphs with time complexity O(m) is obtained by constructive method. The number of corresponding colors is at most 2 or 3 more than the minimum number ...After a necessary condition is given, 3-rainbow coloring of split graphs with time complexity O(m) is obtained by constructive method. The number of corresponding colors is at most 2 or 3 more than the minimum number of colors needed in a 3-rainbow coloring.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
A k-proper total coloring of G is called adjacent distinguishing if for any two adjacent vertices have different color sets. According to the property of trees, the adjacent vertex distinguishing total chromatic numbe...A k-proper total coloring of G is called adjacent distinguishing if for any two adjacent vertices have different color sets. According to the property of trees, the adjacent vertex distinguishing total chromatic number will be determined for the Mycielski graphs of trees using the method of induction.展开更多
A majority k-coloring of a digraph D with k colors is an assignment c:V(D)→{1,2,…,k},such that for every v∈V(D),we have c(w)=c(v)for at most half of all out-neighbors w∈N^(+)(v).For a natural number k≥2,a 1/k-maj...A majority k-coloring of a digraph D with k colors is an assignment c:V(D)→{1,2,…,k},such that for every v∈V(D),we have c(w)=c(v)for at most half of all out-neighbors w∈N^(+)(v).For a natural number k≥2,a 1/k-majority coloring of a digraph is a coloring of the vertices such that each vertex receives the same color as at most a 1/k proportion of its out-neighbours.Kreutzer,Oum,Seymour,van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring.Gireao,Kittipassorn and Popielarz proved that every digraph has a 1/k-majority 2k-coloring and conjectured that every digraph admits a 1/k majority(2k-1)-coloring.We showed that every r-regular digraph D with r>36ln(2n)has a majority 3-coloring and proved that every digraph D with minimum outdegreeδ+>2k2(2k-1)/(k-1)^(2)ln2(n)[(2k-1)n]has a 1/k-majority(2k-1)-coloring.We showed that every r-regular digraph D with r>36ln(2n)has a majority 3-coloring and proved that every digraph D with minimum outdegreeδ+>,2k^(2)(2k-1)^(2)/(k-1)^(2)ln[(2k-1)n]has a 1/k-majority(2k-1)-coloring.And we also proved that every r-regular digraph D with r>3k^(2)(2k-1)/(k-1)^2ln(2n)has a 1/k-majority(2k-1)-coloring.展开更多
We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result a...We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].展开更多
This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, usin...This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, using the matrix semi-tensor product, the robust graph coloring is expressed into a kind of optimization problem taking in an algebraic form of matrices, based on which an algorithm is designed to find all the most robust coloring schemes for any simple graph. Second, an equivalent problem of robust graph coloring is studied, and a necessary and sufficient condition is proposed, from which a new algorithm to find all the most robust coloring schemes is established. Third, a kind of examination timetabling is discussed by using the obtained results, and a method to design a practicable timetabling scheme is presented. Finally, the effectiveness of the results/algorithms presented in this paper is shown by two illustrative examples.展开更多
With the development of Internet technology and human computing, the computing environment has changed dramatically over the last three decades. Cloud computing emerges as a paradigm of Internet computing in which dyn...With the development of Internet technology and human computing, the computing environment has changed dramatically over the last three decades. Cloud computing emerges as a paradigm of Internet computing in which dynamical, scalable and often virtuMized resources are provided as services. With virtualization technology, cloud computing offers diverse services （such as virtual computing, virtual storage, virtual bandwidth, etc.） for the public by means of multi-tenancy mode. Although users are enjoying the capabilities of super-computing and mass storage supplied by cloud computing, cloud security still remains as a hot spot problem, which is in essence the trust management between data owners and storage service providers. In this paper, we propose a data coloring method based on cloud watermarking to recognize and ensure mutual reputations. The experimental results show that the robustness of reverse cloud generator can guarantee users＇ embedded social reputation identifications. Hence, our work provides a reference solution to the critical problem of cloud security.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12271210)the Scientific Research Foundation of Jimei University(Grant No.Q202201).
文摘A t-tone coloring of a graph assigns t distinct colors to each vertex with vertices at distance d having fewer than d colors in common.The t-tone chromatic number of a graph is the smallest number of colors used in all t-tone colorings of that graph.In this article,we study t-tone coloring of some finite planar lattices and obtain exact formulas for their t-tone chromatic number.
基金supported by the Key Technologies R&D Program of Tianjin(Nos.24YFZCSN00030 and 24YFYSHZ00090)。
文摘Image coloring is an inherently uncertain and multimodal problem.By inputting a grayscale image into a coloring network,visually plausible colored photos can be generated.Conventional methods primarily rely on semantic information for image colorization.These methods still suffer from color contamination and semantic confusion.This is largely due to the limited capacity of convolutional neural networks to learn deep semantic information inherent in images effectively.In this paper,we propose a network structure that addresses these limitations by leveraging multi-level semantic information classification and fusion.Additionally,we introduce a global semantic fusion network to combat the issues of color contamination.The proposed coloring encoder accurately extracts object-level semantic information from images.To further enhance visual plausibility,we employ a self-supervised adversarial training method.We train the network structure on various datasets with varying amounts of data and evaluate its performance using the ImageNet validation set and COCO validation set.Experimental results demonstrate that our proposed algorithm can generate more realistic images compared to previous approaches,showcasing its high generalization ability.
基金Supported by the Youth Fund of Lanzhou Jiaotong University(1200061328)。
文摘A proper conflict-free k-coloring of a graph is a proper k-coloring in which each nonisolated vertex has a color that appears ex-actly once in its open neighborhood.A graph is PCF k-colorable if it admits a proper conflict-free k-coloring.The PCF chromatic number of a graph G,denoted by χ_(pcf)(G),is the minimum k such that G is PCF k-colorable.Caro et al conjectured that for a connected graph G with maximum degreeΔ≥3,χ_(pcf)(G)≤Δ+1.One case in this conjecture,a connected graph with maximum degree 3 is PCF 4-colorable,can be derived from the result of Liu and Yu.Jiménez et al stated that the upper bound of PCF chromatic number of a graph G is max{5,x(G)}without a proof.In this paper,we give new proofs of the two results above and derive that for a connected graph G with maximum degreeΔ≥3,its complete subdivision is PCF(Δ+1)-colorable.
文摘A proper edge t-coloring of a graph G is a coloring of its edges with colors 1, 2,..., t, such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each vertex, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. In this paper, we provide a new proof of the result on the colors in cyclically interval edge colorings of simple cycles which was first proved by Rafayel R. Kamalian in the paper “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles, Open Journal of Discrete Mathematics, 2013, 43-48”.
文摘Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C（x） denote the set of colors of vertex x and of the edges incident with x, we call C（x） the color set of x. If C（u） ≠ C（v） for any two different vertices u and v of V （G）, then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.
文摘Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct vertices u and v of V（G）, then f is called a vertex- distinguishing total k-coloring of G. The minimum number k for which there exists a vertex- distinguishing total k-coloring of G is called the vertex-distinguishing total chromatic number of G and denoted by Xvt（G）. The vertex-disjoint union of two cycles of length n is denoted by 2Cn. We will obtain Xvt（2Cn） in this paper.
基金Supported by the NNSF of China(61163037,61163054)Supported by the Scientific Research Foundation of Ningxia University((E):ndzr09-15)
文摘Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 6116303761163054)+2 种基金the Scientific Research Project of Northwest Normal University (No. nwnu-kjcxgc-03-61)the Natural Foudation Project of Ningxia (No. NZ1154)the Scientific Research Foudation Project of Ningxia University (No. (E):ndzr10-7)
文摘Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C（u） =fi C（v） for any two different vertices u and v of V（G）, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The ie iV., minimum number of colors required for a VDIET coloring of G is denoted by X,t[ 1, and it is called the VDIET chromatic number of G. We will give VDIET chromatic numbers for complete bipartite graph K4,n（n ≥ 4）, Kn,n （5 ≤ n ≤21） in this article.
基金The NSF(61163037,61163054) of Chinathe Scientific Research Project(nwnu-kjcxgc-03-61) of Northwest Normal University
文摘Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C（u）=C（v） for any two different vertices u and v of V （G）, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt（G）, and is called the VDIET chromatic number of G. We get the VDIET chromatic numbers of cycles and wheels, and propose related conjectures in this paper.
基金supported by NSFC of China (No. 19871036 and No. 40301037)Faculty Research Grant,Hong Kong Baptist University
文摘A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv∈ E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by X＇Aa（G）, is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.
基金Supported by the National Natural Science Foundation of China（61163037, 61163054, 11261046, 61363060）
文摘Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.
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(11761064)
文摘We give the optimal I-(VI-)total colorings of mC_(4)which are vertex-distinguished by multiple sets by the use of the method of constructing a matrix whose entries are the suitable multiple sets or empty sets and the method of distributing color set in advance.Thereby we obtain I-(VI-)total chromatic numbers of mC_(4)which are vertex-distinguished by multiple sets.
基金Supported by the National Natural Science Foundation of China(No.11001196)
文摘After a necessary condition is given, 3-rainbow coloring of split graphs with time complexity O(m) is obtained by constructive method. The number of corresponding colors is at most 2 or 3 more than the minimum number of colors needed in a 3-rainbow coloring.
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Foundation item： Supported by Natural Science Foundation of China（60503002）
文摘A k-proper total coloring of G is called adjacent distinguishing if for any two adjacent vertices have different color sets. According to the property of trees, the adjacent vertex distinguishing total chromatic number will be determined for the Mycielski graphs of trees using the method of induction.
基金Supported by the National Natural Science Foundation of China(Grant No.12071351)the Natural Science Foundation of Shandong Provence(Grant No.ZR2020MA043).
文摘A majority k-coloring of a digraph D with k colors is an assignment c:V(D)→{1,2,…,k},such that for every v∈V(D),we have c(w)=c(v)for at most half of all out-neighbors w∈N^(+)(v).For a natural number k≥2,a 1/k-majority coloring of a digraph is a coloring of the vertices such that each vertex receives the same color as at most a 1/k proportion of its out-neighbours.Kreutzer,Oum,Seymour,van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring.Gireao,Kittipassorn and Popielarz proved that every digraph has a 1/k-majority 2k-coloring and conjectured that every digraph admits a 1/k majority(2k-1)-coloring.We showed that every r-regular digraph D with r>36ln(2n)has a majority 3-coloring and proved that every digraph D with minimum outdegreeδ+>2k2(2k-1)/(k-1)^(2)ln2(n)[(2k-1)n]has a 1/k-majority(2k-1)-coloring.We showed that every r-regular digraph D with r>36ln(2n)has a majority 3-coloring and proved that every digraph D with minimum outdegreeδ+>,2k^(2)(2k-1)^(2)/(k-1)^(2)ln[(2k-1)n]has a 1/k-majority(2k-1)-coloring.And we also proved that every r-regular digraph D with r>3k^(2)(2k-1)/(k-1)^2ln(2n)has a 1/k-majority(2k-1)-coloring.
文摘We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].
基金This work was supported by the National Natural Science Foundation of China （Nos. G61374065, G61034007, G61374002） the Fund for the Taishan Scholar Project of Shandong Province, the Natural Science Foundation of Shandong Province （No. ZR2010FM013） the Scientific Research and Development Project of Shandong Provincial Education Department （No. J11LA01 ）
文摘This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, using the matrix semi-tensor product, the robust graph coloring is expressed into a kind of optimization problem taking in an algebraic form of matrices, based on which an algorithm is designed to find all the most robust coloring schemes for any simple graph. Second, an equivalent problem of robust graph coloring is studied, and a necessary and sufficient condition is proposed, from which a new algorithm to find all the most robust coloring schemes is established. Third, a kind of examination timetabling is discussed by using the obtained results, and a method to design a practicable timetabling scheme is presented. Finally, the effectiveness of the results/algorithms presented in this paper is shown by two illustrative examples.
基金supported by National Basic Research Program of China (973 Program) (No. 2007CB310800)China Postdoctoral Science Foundation (No. 20090460107 and No. 201003794)
文摘With the development of Internet technology and human computing, the computing environment has changed dramatically over the last three decades. Cloud computing emerges as a paradigm of Internet computing in which dynamical, scalable and often virtuMized resources are provided as services. With virtualization technology, cloud computing offers diverse services （such as virtual computing, virtual storage, virtual bandwidth, etc.） for the public by means of multi-tenancy mode. Although users are enjoying the capabilities of super-computing and mass storage supplied by cloud computing, cloud security still remains as a hot spot problem, which is in essence the trust management between data owners and storage service providers. In this paper, we propose a data coloring method based on cloud watermarking to recognize and ensure mutual reputations. The experimental results show that the robustness of reverse cloud generator can guarantee users＇ embedded social reputation identifications. Hence, our work provides a reference solution to the critical problem of cloud security.
