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
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. 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
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(V, E) be a simple connected graph and k be positive integers. A mapping f from V∪E to {1, 2, ··· , k} is called an adjacent vertex-distinguishing E-total coloring of G(abbreviated to k-AVDETC), i...Let G(V, E) be a simple connected graph and k be positive integers. A mapping f from V∪E to {1, 2, ··· , k} is called an adjacent vertex-distinguishing E-total coloring of G(abbreviated to k-AVDETC), if for uv ∈ E(G), we have f(u) ≠ f(v), f(u) ≠ f(uv), f(v) ≠ f(uv), C(u) ≠C(v), where C(u) = {f(u)}∪{f(uv)|uv ∈ E(G)}. The least number of k colors required for which G admits a k-coloring is called the adjacent vertex-distinguishing E-total chromatic number of G is denoted by x^e_(at) (G). In this paper, the adjacent vertexdistinguishing E-total colorings of some join graphs C_m∨G_n are obtained, where G_n is one of a star S_n , a fan F_n , a wheel W_n and a complete graph K_n . As a consequence, the adjacent vertex-distinguishing E-total chromatic numbers of C_m∨G_n are confirmed.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
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
Let f be a proper edge coloring of G using k colors.For each x∈V(G),the set of the colors appearing on the edges incident with x is denoted by S_f(x)or simply S(x)if no confusion arise.If S(u)■S(v)and S(v)■S(u)for ...Let f be a proper edge coloring of G using k colors.For each x∈V(G),the set of the colors appearing on the edges incident with x is denoted by S_f(x)or simply S(x)if no confusion arise.If S(u)■S(v)and S(v)■S(u)for any two adjacent vertices u and v,then f is called a Smarandachely adjacent vertex distinguishing proper edge coloring using k colors,or k-SA-edge coloring.The minimum number k for which G has a Smarandachely adjacent-vertex-distinguishing proper edge coloring using k colors is called the Smarandachely adjacent-vertex-distinguishing proper edge chromatic number,or SAedge chromatic number for short,and denoted byχ'_(sa)(G).In this paper,we have discussed the SA-edge chromatic number of K_4∨K_n.展开更多
Let G be a simple graph with no isolated edge. An/-total coloring of a graphG is a mapping Ф : V(G) U E(G) → (1, 2,…… , k) such that no adjacent vertices receive thesame color and no adjacent edges receive ...Let G be a simple graph with no isolated edge. An/-total coloring of a graphG is a mapping Ф : V(G) U E(G) → (1, 2,…… , k) such that no adjacent vertices receive thesame color and no adjacent edges receive the same color. An/-total coloring of a graph G issaid to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, wehave CФ(u) ≠ CФ(v), where CФ(u) denotes the set of colors of u and its incident edges. Theminimum number of colors required for an adjacent vertex distinguishing I-total coloring of GG is called the adjacent vertex distinguishing I-total chromatic number, denoted by Xat(G).In this paper, we characterize the adjacent vertex distinguishing I-total chromatic numberof outerplanar graphs.展开更多
A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge ...A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph C is denoted by Xs'8(G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with △(G) ≥ 4, respectively.展开更多
IEEE 802.11 based wireless mesh networks with directional antennas are expected to be a new promising technology and an economic approach for providing wireless broadband services in rural areas.In this paper,we discu...IEEE 802.11 based wireless mesh networks with directional antennas are expected to be a new promising technology and an economic approach for providing wireless broadband services in rural areas.In this paper,we discuss interference models and address how they can affect the design of channel assignment in rural mesh networks.We present a new channel assignment framework based on graph coloring for rural wireless mesh networks.The goal of the framework is to allow synchronously transmitting or receiving data from multiple neighbor links at the same time,and continuously doing full-duplex data transfer on every link,creating an efficient rural mesh network without interference.Channel assignment is shown to be NP-hard.We frame this channel allocation problem in terms of Adjacent Vertex Distinguishing Edge Coloring(AVDEC).Detailed assignment results on grid topology are presented and discussed.Furthermore,we design an algorithm.Finally,we evaluate the performance of the proposed algorithm through extensive simulations and show the algorithm is effective to the regular grid topologies,and the number of colors used by the algorithm is upper bounded by+1.Hence the algorithm guarantees that the number of channels available in standards such as IEEE802.11a is sufficient to have a valid AVDEC for many grid topologies.We also evaluate the proposed algorithm for arbitrary graphs.The algorithm provides a lower upper bound on the minimum number of channels to the AVDEC index channel assignment problem.展开更多
A proper <em>k</em>-edge coloring of a graph <em>G</em> = (<em>V</em>(<em>G</em>), <em>E</em>(<em>G</em>)) is an assignment <em>c</em>...A proper <em>k</em>-edge coloring of a graph <em>G</em> = (<em>V</em>(<em>G</em>), <em>E</em>(<em>G</em>)) is an assignment <em>c</em>: <em>E</em>(<em>G</em>) → {1, 2, …, <em>k</em>} such that no two adjacent edges receive the same color. A neighbor sum distinguishing <em>k</em>-edge coloring of <em>G</em> is a proper <em>k</em>-edge coloring of <em>G</em> such that <img src="Edit_28f0a24c-7d3f-4bdc-b58c-46dfa2add4b4.bmp" alt="" /> for each edge <em>uv</em> ∈ <em>E</em>(<em>G</em>). The neighbor sum distinguishing index of a graph <em>G</em> is the least integer <em>k</em> such that <em>G </em>has such a coloring, denoted by <em>χ’</em><sub>Σ</sub>(<em>G</em>). Let <img src="Edit_7525056f-b99d-4e38-b940-618d16c061e2.bmp" alt="" /> be the maximum average degree of <em>G</em>. In this paper, we prove <em>χ</em>’<sub>Σ</sub>(<em>G</em>) ≤ max{9, Δ(<em>G</em>) +1} for any normal graph <em>G</em> with <img src="Edit_e28e38d5-9b6d-46da-bfce-2aae47cc36f3.bmp" alt="" />. Our approach is based on the discharging method and Combinatorial Nullstellensatz.展开更多
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 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.
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.
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 NNSF of China(10771091)Supported by the Qinglan Project of Lianyungang Teacher’s College(2009QLD3)
文摘Let G(V, E) be a simple connected graph and k be positive integers. A mapping f from V∪E to {1, 2, ··· , k} is called an adjacent vertex-distinguishing E-total coloring of G(abbreviated to k-AVDETC), if for uv ∈ E(G), we have f(u) ≠ f(v), f(u) ≠ f(uv), f(v) ≠ f(uv), C(u) ≠C(v), where C(u) = {f(u)}∪{f(uv)|uv ∈ E(G)}. The least number of k colors required for which G admits a k-coloring is called the adjacent vertex-distinguishing E-total chromatic number of G is denoted by x^e_(at) (G). In this paper, the adjacent vertexdistinguishing E-total colorings of some join graphs C_m∨G_n are obtained, where G_n is one of a star S_n , a fan F_n , a wheel W_n and a complete graph K_n . As a consequence, the adjacent vertex-distinguishing E-total chromatic numbers of C_m∨G_n are confirmed.
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 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 NNSF of China(61163037,61163054,61363060)
文摘Let f be a proper edge coloring of G using k colors.For each x∈V(G),the set of the colors appearing on the edges incident with x is denoted by S_f(x)or simply S(x)if no confusion arise.If S(u)■S(v)and S(v)■S(u)for any two adjacent vertices u and v,then f is called a Smarandachely adjacent vertex distinguishing proper edge coloring using k colors,or k-SA-edge coloring.The minimum number k for which G has a Smarandachely adjacent-vertex-distinguishing proper edge coloring using k colors is called the Smarandachely adjacent-vertex-distinguishing proper edge chromatic number,or SAedge chromatic number for short,and denoted byχ'_(sa)(G).In this paper,we have discussed the SA-edge chromatic number of K_4∨K_n.
基金Supported by the National Natural Science Foundation of China(61163037,61163054,61363060)
文摘Let G be a simple graph with no isolated edge. An/-total coloring of a graphG is a mapping Ф : V(G) U E(G) → (1, 2,…… , k) such that no adjacent vertices receive thesame color and no adjacent edges receive the same color. An/-total coloring of a graph G issaid to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, wehave CФ(u) ≠ CФ(v), where CФ(u) denotes the set of colors of u and its incident edges. Theminimum number of colors required for an adjacent vertex distinguishing I-total coloring of GG is called the adjacent vertex distinguishing I-total chromatic number, denoted by Xat(G).In this paper, we characterize the adjacent vertex distinguishing I-total chromatic numberof outerplanar graphs.
基金Supported by the National Natural Science Foundation of China(10971198)the Zhejiang Natural Science Foundation of China(Z6110786)
文摘A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph C is denoted by Xs'8(G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with △(G) ≥ 4, respectively.
基金Supported by the National Natural Science Foundation of China(No.71231004 and No.61004086)
文摘IEEE 802.11 based wireless mesh networks with directional antennas are expected to be a new promising technology and an economic approach for providing wireless broadband services in rural areas.In this paper,we discuss interference models and address how they can affect the design of channel assignment in rural mesh networks.We present a new channel assignment framework based on graph coloring for rural wireless mesh networks.The goal of the framework is to allow synchronously transmitting or receiving data from multiple neighbor links at the same time,and continuously doing full-duplex data transfer on every link,creating an efficient rural mesh network without interference.Channel assignment is shown to be NP-hard.We frame this channel allocation problem in terms of Adjacent Vertex Distinguishing Edge Coloring(AVDEC).Detailed assignment results on grid topology are presented and discussed.Furthermore,we design an algorithm.Finally,we evaluate the performance of the proposed algorithm through extensive simulations and show the algorithm is effective to the regular grid topologies,and the number of colors used by the algorithm is upper bounded by+1.Hence the algorithm guarantees that the number of channels available in standards such as IEEE802.11a is sufficient to have a valid AVDEC for many grid topologies.We also evaluate the proposed algorithm for arbitrary graphs.The algorithm provides a lower upper bound on the minimum number of channels to the AVDEC index channel assignment problem.
文摘A proper <em>k</em>-edge coloring of a graph <em>G</em> = (<em>V</em>(<em>G</em>), <em>E</em>(<em>G</em>)) is an assignment <em>c</em>: <em>E</em>(<em>G</em>) → {1, 2, …, <em>k</em>} such that no two adjacent edges receive the same color. A neighbor sum distinguishing <em>k</em>-edge coloring of <em>G</em> is a proper <em>k</em>-edge coloring of <em>G</em> such that <img src="Edit_28f0a24c-7d3f-4bdc-b58c-46dfa2add4b4.bmp" alt="" /> for each edge <em>uv</em> ∈ <em>E</em>(<em>G</em>). The neighbor sum distinguishing index of a graph <em>G</em> is the least integer <em>k</em> such that <em>G </em>has such a coloring, denoted by <em>χ’</em><sub>Σ</sub>(<em>G</em>). Let <img src="Edit_7525056f-b99d-4e38-b940-618d16c061e2.bmp" alt="" /> be the maximum average degree of <em>G</em>. In this paper, we prove <em>χ</em>’<sub>Σ</sub>(<em>G</em>) ≤ max{9, Δ(<em>G</em>) +1} for any normal graph <em>G</em> with <img src="Edit_e28e38d5-9b6d-46da-bfce-2aae47cc36f3.bmp" alt="" />. Our approach is based on the discharging method and Combinatorial Nullstellensatz.