An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u...An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u）-f（v）｜≥1 if dG（u,v） = 3. The L（3, 2,1）-labeling problem is to find the smallest number λ3（G） such that there exists an L（3, 2,1）-labeling function with no label greater than it. This paper studies the problem for bipartite graphs. We obtain some bounds of λ3 for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree T such that λ3（T） attains the minimum value.展开更多
L (2, 1)-labeling number, λ(G( Z , D)) , of distance graph G( Z , D) is studied. For general finite distance set D , it is shown that 2D+2≤λ(G( Z , D))≤D 2+3D. Furthermore, λ(G( Z , D)) ≤8 when...L (2, 1)-labeling number, λ(G( Z , D)) , of distance graph G( Z , D) is studied. For general finite distance set D , it is shown that 2D+2≤λ(G( Z , D))≤D 2+3D. Furthermore, λ(G( Z , D)) ≤8 when D consists of two prime positive odd integers is proved. Finally, a new concept to study the upper bounds of λ(G) for some special D is introduced. For these sets, the upper bound is improved to 7.展开更多
A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this pape...A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.展开更多
The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for ...The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for any fixed positive integer k. Suppose k, a ∈ N and k,a≥2. If k≥a, then λ({a, a + 1,..., a + k - 1}) = 2(a + k-1). Otherwise, λ({a, a + 1, ..., a + k- 1}) ≤min{2(a + k-1), 6k -2}. When D consists of two positive integers,6≤λ(D)≤8. For thespecial distance sets D = {k, k + 1}(any k ∈N), the upper bound of λ(D) is improved to 7.展开更多
An L（2, 1）-labelling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that │f（u） - f（v）│≥2 if dG（u, v） = 1 and │f（u） - f（v）│ ≥ 1 if dG（u, v） = 2. Th...An L（2, 1）-labelling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that │f（u） - f（v）│≥2 if dG（u, v） = 1 and │f（u） - f（v）│ ≥ 1 if dG（u, v） = 2. The L（2, 1）-labelling problem is to find the smallest number, denoted by A（G）, such that there exists an L（2, 1）-labelling function with no label greater than it. In this paper, we study this problem for trees. Our results improve the result of Wang [The L（2, 1）-labelling of trees, Discrete Appl. Math. 154 （2006） 598-603].展开更多
An L(h,k)-labeling of a graph G is an assignment of non-negative integers to the vertices such that if two vertices u and v are adjacent then they receive labels that differ by at least h, and when u and v are not adj...An L(h,k)-labeling of a graph G is an assignment of non-negative integers to the vertices such that if two vertices u and v are adjacent then they receive labels that differ by at least h, and when u and v are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least k. The span λ of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let λ<sub>h</sub>k</sup> ( G )denote the least λ such that G admits an L(h,k) -labeling using labels from {0,1,...λ}. A Cayley graph of group is called circulant graph of order n, if the group is isomorphic to Z<sub>n.</sub> In this paper, initially we investigate the L(h,k) -labeling for circulant graphs with “large” connection sets, and then we extend our observation and find the span of L(h,k) -labeling for any circulants of order n. .展开更多
Given a graph G and a positive integer d, an L( d, 1) -labeling of G is afunction / that assigns to each vertex of G a non-negative integer such that |f(u)-f (v) | >=d ifd_c(u, v) =1;|f(u)-f(v) | >=1 if d_c(u, v...Given a graph G and a positive integer d, an L( d, 1) -labeling of G is afunction / that assigns to each vertex of G a non-negative integer such that |f(u)-f (v) | >=d ifd_c(u, v) =1;|f(u)-f(v) | >=1 if d_c(u, v) =2. The L(d, 1)-labeling number of G, lambda_d(G) is theminimum range span of labels over all such labelings, which is motivated by the channel assignmentproblem. We consider the question of finding the minimum edge span beta_d( G) of this labeling.Several classes of graphs such as cycles, trees, complete k-partite graphs, chordal graphs includingtriangular lattice and square lattice which are important to a telecommunication problem arestudied, and exact values are given.展开更多
An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G...An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L（d1,d2,...,dt）-number λ（G;d1,d2,... ,dt） of G is the smallest integer number k such that G has an L（d1,d2,...,dr）- labeling with max{f （x）｜x ∈ V（G）} = k. In this paper, we obtain the exact values for λ（Cn; 2, 2, 1） and λ（Cn; 3, 2, 1）, and present lower and upper bounds for λ（Cn; 2,..., 2, 1,..., 1）展开更多
Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are...Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.展开更多
An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference betw...An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G.展开更多
An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The ...An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The L（2, 1）-labeling number λ（G） of G is the smallest number k such that G has an L（2, 1）-labeling with max{f（v） ： v ∈ V（G）} = k. We study the L（3, 2, 1）-labeling which is a generalization of the L（2, 1）-labeling on the graph formed by the （Cartesian） product and composition of 3 graphs and derive the upper bounds of λ3（G） of the graph.展开更多
L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assig...L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assignment of integers to the vertices of G such that adjacent vertices receive integers which differ by at least s, and vertices that are at distance of two receive integers which differ by at least t. Given an L（s, t） -labeling f of a graph G, the L（s, t） edge span of f, βst （ G, f） = max { ｜f（u） -f（v）｜： （ u, v） ∈ E（G） } is defined. The L（ s, t） edge span of G, βst（G）, is minβst（G,f）, where the minimum runs over all L（s, t）-labelings f of G. Let T be any tree with a maximum degree of △≥2. It is proved that if 2s≥t≥0, then βst（T） =（ [△/2 ] - 1）t ＋s; if 0≤2s 〈 t and △ is even, then βst（T） = [ （△ - 1） t/2 ] ; and if 0 ≤2s 〈 t and △ is odd, then βst（T） = （△ - 1） t/2 ＋ s. Thus, the L（s, t） edge spans of the Cartesian product of two paths and of the square lattice are completely determined.展开更多
An L（j, k）-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart, and vertices at distance two receive integers w...An L（j, k）-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart, and vertices at distance two receive integers which are at least k apart. Given an L（j, k）-labeling f of G, define the L（j, k） edge span of f, βj,k（G,f） =max{ ｜f（x）-f（y）｜： {x,y}∈E（G）}. The L（j,k） edge span of G, βj,k （G） is min βj,k（ G, f）, where the minimum runs over all L（j, k）-labelings f of G. The real L（.j, k）-labeling of a graph G is a generalization of the L（j, k）-labeling. It is an assignment of nonnegative real numbers to the vertices of G satisfying the same distance one and distance two conditions. The real L（j, k） edge span of a graph G is defined accordingly, and is denoted by βj,k（G）. This paper investigates some properties of the L（j, k） edge span and the real L（j, k） edge span of graphs, and completely determines the edge spans of cycles and complete t-partite graphs.展开更多
基金The NSF (60673048) of China the NSF (KJ2009B002,KJ2009B237Z) of Education Ministry of Anhui Province.
文摘An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u）-f（v）｜≥1 if dG（u,v） = 3. The L（3, 2,1）-labeling problem is to find the smallest number λ3（G） such that there exists an L（3, 2,1）-labeling function with no label greater than it. This paper studies the problem for bipartite graphs. We obtain some bounds of λ3 for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree T such that λ3（T） attains the minimum value.
文摘L (2, 1)-labeling number, λ(G( Z , D)) , of distance graph G( Z , D) is studied. For general finite distance set D , it is shown that 2D+2≤λ(G( Z , D))≤D 2+3D. Furthermore, λ(G( Z , D)) ≤8 when D consists of two prime positive odd integers is proved. Finally, a new concept to study the upper bounds of λ(G) for some special D is introduced. For these sets, the upper bound is improved to 7.
基金National Natural Science Foundation of China(No.10671074 and No.60673048)Natural Science Foundation of Education Ministry of Anhui Province(No.KJ2007B124 and No.2006KJ256B)
文摘A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.
基金Supported by the Natural Science Foundation of Education Ministry of Anhui Province (No.KJ2010B138)the Foundation for the Excellent Young Talents of Anhui Province(No.2010SQRL136ZD)the Natural Science Foundation of Chuzhou University(No.2008kj013B)
文摘The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for any fixed positive integer k. Suppose k, a ∈ N and k,a≥2. If k≥a, then λ({a, a + 1,..., a + k - 1}) = 2(a + k-1). Otherwise, λ({a, a + 1, ..., a + k- 1}) ≤min{2(a + k-1), 6k -2}. When D consists of two positive integers,6≤λ(D)≤8. For thespecial distance sets D = {k, k + 1}(any k ∈N), the upper bound of λ(D) is improved to 7.
基金Supported by the National Natural Science Foundation of China (No. 10971248,11101057)Anhui Provincial Natural Science Foundation (No. 10040606Q45)Postdoctoral Science Foundation of Jiangsu Provinc (No.1102095C)
文摘An L（2, 1）-labelling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that │f（u） - f（v）│≥2 if dG（u, v） = 1 and │f（u） - f（v）│ ≥ 1 if dG（u, v） = 2. The L（2, 1）-labelling problem is to find the smallest number, denoted by A（G）, such that there exists an L（2, 1）-labelling function with no label greater than it. In this paper, we study this problem for trees. Our results improve the result of Wang [The L（2, 1）-labelling of trees, Discrete Appl. Math. 154 （2006） 598-603].
文摘An L(h,k)-labeling of a graph G is an assignment of non-negative integers to the vertices such that if two vertices u and v are adjacent then they receive labels that differ by at least h, and when u and v are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least k. The span λ of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let λ<sub>h</sub>k</sup> ( G )denote the least λ such that G admits an L(h,k) -labeling using labels from {0,1,...λ}. A Cayley graph of group is called circulant graph of order n, if the group is isomorphic to Z<sub>n.</sub> In this paper, initially we investigate the L(h,k) -labeling for circulant graphs with “large” connection sets, and then we extend our observation and find the span of L(h,k) -labeling for any circulants of order n. .
文摘Given a graph G and a positive integer d, an L( d, 1) -labeling of G is afunction / that assigns to each vertex of G a non-negative integer such that |f(u)-f (v) | >=d ifd_c(u, v) =1;|f(u)-f(v) | >=1 if d_c(u, v) =2. The L(d, 1)-labeling number of G, lambda_d(G) is theminimum range span of labels over all such labelings, which is motivated by the channel assignmentproblem. We consider the question of finding the minimum edge span beta_d( G) of this labeling.Several classes of graphs such as cycles, trees, complete k-partite graphs, chordal graphs includingtriangular lattice and square lattice which are important to a telecommunication problem arestudied, and exact values are given.
基金the National Natural Science Foundation of China (No.10531070)the National Basic Research Program of China 973 Program (No.2006AA11Z209)the Natural Science Foundation of Shanghai City (No.06ZR14049)
文摘An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L（d1,d2,...,dt）-number λ（G;d1,d2,... ,dt） of G is the smallest integer number k such that G has an L（d1,d2,...,dr）- labeling with max{f （x）｜x ∈ V（G）} = k. In this paper, we obtain the exact values for λ（Cn; 2, 2, 1） and λ（Cn; 3, 2, 1）, and present lower and upper bounds for λ（Cn; 2,..., 2, 1,..., 1）
基金The National Natural Science Foundation of China(No.10971025)
文摘Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.
文摘An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G.
文摘An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The L（2, 1）-labeling number λ（G） of G is the smallest number k such that G has an L（2, 1）-labeling with max{f（v） ： v ∈ V（G）} = k. We study the L（3, 2, 1）-labeling which is a generalization of the L（2, 1）-labeling on the graph formed by the （Cartesian） product and composition of 3 graphs and derive the upper bounds of λ3（G） of the graph.
基金The National Natural Science Foundation of China(No10671033)Southeast University Science Foundation ( NoXJ0607230)
文摘L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assignment of integers to the vertices of G such that adjacent vertices receive integers which differ by at least s, and vertices that are at distance of two receive integers which differ by at least t. Given an L（s, t） -labeling f of a graph G, the L（s, t） edge span of f, βst （ G, f） = max { ｜f（u） -f（v）｜： （ u, v） ∈ E（G） } is defined. The L（ s, t） edge span of G, βst（G）, is minβst（G,f）, where the minimum runs over all L（s, t）-labelings f of G. Let T be any tree with a maximum degree of △≥2. It is proved that if 2s≥t≥0, then βst（T） =（ [△/2 ] - 1）t ＋s; if 0≤2s 〈 t and △ is even, then βst（T） = [ （△ - 1） t/2 ] ; and if 0 ≤2s 〈 t and △ is odd, then βst（T） = （△ - 1） t/2 ＋ s. Thus, the L（s, t） edge spans of the Cartesian product of two paths and of the square lattice are completely determined.
基金The National Natural Science Foundation of China (No10971025)
文摘An L（j, k）-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart, and vertices at distance two receive integers which are at least k apart. Given an L（j, k）-labeling f of G, define the L（j, k） edge span of f, βj,k（G,f） =max{ ｜f（x）-f（y）｜： {x,y}∈E（G）}. The L（j,k） edge span of G, βj,k （G） is min βj,k（ G, f）, where the minimum runs over all L（j, k）-labelings f of G. The real L（.j, k）-labeling of a graph G is a generalization of the L（j, k）-labeling. It is an assignment of nonnegative real numbers to the vertices of G satisfying the same distance one and distance two conditions. The real L（j, k） edge span of a graph G is defined accordingly, and is denoted by βj,k（G）. This paper investigates some properties of the L（j, k） edge span and the real L（j, k） edge span of graphs, and completely determines the edge spans of cycles and complete t-partite graphs.
