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 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
This paper presents a robust filter called the quaternion Hardy filter(QHF)for color image edge detection.The QHF can be capable of color edge feature enhancement and noise resistance.QHF can be used flexibly by selec...This paper presents a robust filter called the quaternion Hardy filter(QHF)for color image edge detection.The QHF can be capable of color edge feature enhancement and noise resistance.QHF can be used flexibly by selecting suitable parameters to handle different levels of noise.In particular,the quaternion analytic signal,which is an effective tool in color image processing,can also be produced by quaternion Hardy filtering with specific parameters.Based on the QHF and the improved Di Zenzo gradient operator,a novel color edge detection algorithm is proposed;importantly,it can be efficiently implemented by using the fast discrete quaternion Fourier transform technique.From the experimental results,we conclude that the minimum PSNR improvement rate is 2.3%and the minimum SSIM improvement rate is 30.2%on the CSEE database.The experiments demonstrate that the proposed algorithm outperforms several widely used algorithms.展开更多
In this paper,a new type of edge coloring of graphs together with an algorithm for such an edge coloring is presented to construct some columnweight three low-density parity-check(LDPC)codes whose Tanner graphs are fr...In this paper,a new type of edge coloring of graphs together with an algorithm for such an edge coloring is presented to construct some columnweight three low-density parity-check(LDPC)codes whose Tanner graphs are free of 4-cycles.This kind of edge coloring is applied on some well-known classes of graphs such as complete graphs and complete bipartite graphs to generate some column-weight 3 LDPC codes having flexibility in terms of code length and rate.Interestingly,the constructed(3;k)-regular codes with regularities k=4;5;:::;22 have lengths n=12;20;26,35;48;57;70;88;104;117;140;155;176;204;228;247;280;301;330;having minimum block length compared to the best known similar codes in the literature.In addition to linear complexity of generating such parity-check matrices,they can be considered as the base matrices of some quasi-cyclic(QC)LDPC codes with maximum achievable girth 18,which inherit the low-complexity encoder implementations of QC-LDPC codes.Simulation results show that the QC-LDPC codes with large girth lifted from the constructed base matrices have good performances and outperform random codes,progressive edge growth LDPC codes,some finite fields and group rings based QC-LDPC codes and also have a close competition to the standard IEEE 802.16e(WiMAX)code.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 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
Let G be a multigraph with vertex set V(G). Assume that a positive integer f(v) with 1 ≤ f(v) ≤ d(v) is associated with each vertex v ∈ V. An edge coloring of G is called an f-edge cover-coloring, if each c...Let G be a multigraph with vertex set V(G). Assume that a positive integer f(v) with 1 ≤ f(v) ≤ d(v) is associated with each vertex v ∈ V. An edge coloring of G is called an f-edge cover-coloring, if each color appears at each vertex v at least f(v) times. Let X'fc(G) be the maximum positive integer k for which an f-edge cover-coloring with k colors of G exists. In this paper, we give a new lower bound of X'fc(G), which is sharp.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
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
The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of the same color.An outerplane graph with independent crossings is a gra...The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of the same color.An outerplane graph with independent crossings is a graph embedded in the plane in such a way that all vertices are on the outer face and two pairs of crossing edges share no common end vertex.It is proved that every outerplane graph with independent crossings and maximum degreeΔhas strong chromatic index at most 4Δ-6 if Δ≥4,and at most 8 ifΔ≤3.Both bounds are sharp.展开更多
An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G.The acyclic chromatic index χ'α(G) of G is the smallest k such that G has an acyclic edge coloring u...An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G.The acyclic chromatic index χ'α(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.It was conjectured that every simple graph G with maximum degree Δ has χ'_α(G) ≤Δ+2.A1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge.In this paper,we show that every 1-planar graph G without 4-cycles has χ'_α(G)≤Δ+22.展开更多
A proper edge k-coloring is a mappingΦ:E(G)-→{1,2,...,k}such that any two adjacent edges receive different colors.A proper edge k-coloringΦof G is called acyclic if there are no bichromatic cycles in G.The acyclic ...A proper edge k-coloring is a mappingΦ:E(G)-→{1,2,...,k}such that any two adjacent edges receive different colors.A proper edge k-coloringΦof G is called acyclic if there are no bichromatic cycles in G.The acyclic chromatic index of G,denoted by Xa(G),is the smallest integer k such that G is acyclically edge k-colorable.In this paper,we show that if G is a plane graph without 4-,6-cycles and intersecting 3-cycles,△(G)≥9,then Xa(G)≤△(G)+1.展开更多
A proper k-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 color set of edges incident to u is not equal to the color set of edges ...A proper k-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 color set of edges incident to u is not equal to the color set of edges incident to v, where uv ∈E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ'αα(G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ'αα(G)≤32△.展开更多
A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each e...A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let X(G ) denote the smallest value k in such a ' G coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) 〈 5 then x'(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.展开更多
Let Ф : E(G)→ {1, 2,…, k}be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if ∑eЭu Ф(e)≠∑eЭu Ф(e) for each edge uv∈E(G).The smallest value k for ...Let Ф : E(G)→ {1, 2,…, k}be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if ∑eЭu Ф(e)≠∑eЭu Ф(e) for each edge uv∈E(G).The smallest value k for which G has such a coloring is denoted by χ'Σ(G) which makes sense for graphs containing no isolated edge(we call such graphs normal). It was conjectured by Flandrin et al. that χ'Σ(G) ≤△(G) + 2 for all normal graphs,except for C5. Let mad(G) = max{(2|E(H)|)/(|V(H)|)|HЭG}be the maximum average degree of G. In this paper,we prove that if G is a normal graph with△(G)≥5 and mad(G) 〈 3-2/(△(G)), then χ'Σ(G)≤△(G) + 1. This improves the previous results and the bound △(G) + 1 is sharp.展开更多
Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, a...Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χ áve (G) of some special graphs and present a conjecture.展开更多
The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χt(G). If the difference of the numbers of any two total independent partition...The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χt(G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one', then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χet(G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order.展开更多
A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3.The star chromatic indexχ'_(st)(G)of a graph G is the smallest integer k such that G h...A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3.The star chromatic indexχ'_(st)(G)of a graph G is the smallest integer k such that G has a star k-edge-coloring.The list star chromatic index ch'st(G)is defined analogously.The star edge coloring problem is known to be NP-complete,and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear.In this paper,we study,in contrast,the best linear upper bound for sparse graph classes.We show that for everyε>0 there exists a constant c(ε)such that if mad(G)<8/3-ε,then■and the coefficient 3/2 ofΔis the best possible.The proof applies a newly developed coloring extension method by assigning color sets with different sizes.展开更多
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 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 in part by the Science and Technology Development Fund,Macao SAR FDCT/085/2018/A2the Guangdong Basic and Applied Basic Research Foundation(2019A1515111185)。
文摘This paper presents a robust filter called the quaternion Hardy filter(QHF)for color image edge detection.The QHF can be capable of color edge feature enhancement and noise resistance.QHF can be used flexibly by selecting suitable parameters to handle different levels of noise.In particular,the quaternion analytic signal,which is an effective tool in color image processing,can also be produced by quaternion Hardy filtering with specific parameters.Based on the QHF and the improved Di Zenzo gradient operator,a novel color edge detection algorithm is proposed;importantly,it can be efficiently implemented by using the fast discrete quaternion Fourier transform technique.From the experimental results,we conclude that the minimum PSNR improvement rate is 2.3%and the minimum SSIM improvement rate is 30.2%on the CSEE database.The experiments demonstrate that the proposed algorithm outperforms several widely used algorithms.
基金The authors would like to thank anonymous referees for their valuable comments enabled us to greatly improve the quality of the paper.The research of the first author is partially supported by Shahrekord University grant No.97GRN1M1465.
文摘In this paper,a new type of edge coloring of graphs together with an algorithm for such an edge coloring is presented to construct some columnweight three low-density parity-check(LDPC)codes whose Tanner graphs are free of 4-cycles.This kind of edge coloring is applied on some well-known classes of graphs such as complete graphs and complete bipartite graphs to generate some column-weight 3 LDPC codes having flexibility in terms of code length and rate.Interestingly,the constructed(3;k)-regular codes with regularities k=4;5;:::;22 have lengths n=12;20;26,35;48;57;70;88;104;117;140;155;176;204;228;247;280;301;330;having minimum block length compared to the best known similar codes in the literature.In addition to linear complexity of generating such parity-check matrices,they can be considered as the base matrices of some quasi-cyclic(QC)LDPC codes with maximum achievable girth 18,which inherit the low-complexity encoder implementations of QC-LDPC codes.Simulation results show that the QC-LDPC codes with large girth lifted from the constructed base matrices have good performances and outperform random codes,progressive edge growth LDPC codes,some finite fields and group rings based QC-LDPC codes and also have a close competition to the standard IEEE 802.16e(WiMAX)code.
基金supported by the National Natural Science Foundation of China (61163054),supported by the National Natural Science Foundation of China (61163037)
文摘It has been known that determining the exact value of vertex distinguishing edge index X '8(G) of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete, graphs, and graphs with maximum degree 2. Let rid(G) denote the number of vertices of degree d in G, and let X'es(G) be the equitable vertex distinguishing edge index of G. We show that a tree T holds nl (T) ≤ X 's (T) ≤ n1 (T) + 1 and X's(T) = X'es(T) if T satisfies one of the following conditions (i) n2(T) ≤△(T) or (ii) there exists a constant c with respect to 0 〈 c 〈 1 such that n2(T) △ cn1(T) and ∑3 ≤d≤△(T)nd(T) ≤ (1 - c)n1(T) + 1.
基金Supported by the National Natural Science Foundation of China(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.
文摘Let G be a multigraph with vertex set V(G). Assume that a positive integer f(v) with 1 ≤ f(v) ≤ d(v) is associated with each vertex v ∈ V. An edge coloring of G is called an f-edge cover-coloring, if each color appears at each vertex v at least f(v) times. Let X'fc(G) be the maximum positive integer k for which an f-edge cover-coloring with k colors of G exists. In this paper, we give a new lower bound of X'fc(G), which is sharp.
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.
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”.
基金supported by the Natural Science Basic Research Plan in Shaanxi Province of China(No.2023-JC-YB-001)the National Natural Science Foundation of China(No.11871055).
文摘The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of the same color.An outerplane graph with independent crossings is a graph embedded in the plane in such a way that all vertices are on the outer face and two pairs of crossing edges share no common end vertex.It is proved that every outerplane graph with independent crossings and maximum degreeΔhas strong chromatic index at most 4Δ-6 if Δ≥4,and at most 8 ifΔ≤3.Both bounds are sharp.
基金Research supported by the National Natural Science Foundation of China (No.12031018)Research supported by the National Natural Science Foundation of China (No.12071048)+3 种基金Research supported by the National Natural Science Foundation of China(No.12071351)Science and Technology Commission of Shanghai Municipality (No.18dz2271000)Doctoral Scientific Research Foundation of Weifang University (No.2021BS01)Natural Science Foundation of Shandong Province (No.ZR2022MA060)。
文摘An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G.The acyclic chromatic index χ'α(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.It was conjectured that every simple graph G with maximum degree Δ has χ'_α(G) ≤Δ+2.A1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge.In this paper,we show that every 1-planar graph G without 4-cycles has χ'_α(G)≤Δ+22.
文摘A proper edge k-coloring is a mappingΦ:E(G)-→{1,2,...,k}such that any two adjacent edges receive different colors.A proper edge k-coloringΦof G is called acyclic if there are no bichromatic cycles in G.The acyclic chromatic index of G,denoted by Xa(G),is the smallest integer k such that G is acyclically edge k-colorable.In this paper,we show that if G is a plane graph without 4-,6-cycles and intersecting 3-cycles,△(G)≥9,then Xa(G)≤△(G)+1.
基金Supported by the Natural Science Foundation of Gansu Province(3ZS051-A25-025)
文摘A proper k-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 color set of edges incident to u is not equal to the color set of edges incident to v, where uv ∈E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ'αα(G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ'αα(G)≤32△.
基金Supported by National Natural Science Foundation of China(Grant Nos.11371355,11471193,11271006,11631014)the Foundation for Distinguished Young Scholars of Shandong Province(Grant No.JQ201501)the Fundamental Research Funds of Shandong University and Independent Innovation Foundation of Shandong University(Grant No.IFYT14012)
文摘A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let X(G ) denote the smallest value k in such a ' G coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) 〈 5 then x'(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.
基金Supported by the National Natural Science Foundation of China(11471193,11631014)the Foundation for Distinguished Young Scholars of Shandong Province(JQ201501)+1 种基金the Fundamental Research Funds of Shandong UniversityIndependent Innovation Foundation of Shandong University(IFYT14012)
文摘Let Ф : E(G)→ {1, 2,…, k}be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if ∑eЭu Ф(e)≠∑eЭu Ф(e) for each edge uv∈E(G).The smallest value k for which G has such a coloring is denoted by χ'Σ(G) which makes sense for graphs containing no isolated edge(we call such graphs normal). It was conjectured by Flandrin et al. that χ'Σ(G) ≤△(G) + 2 for all normal graphs,except for C5. Let mad(G) = max{(2|E(H)|)/(|V(H)|)|HЭG}be the maximum average degree of G. In this paper,we prove that if G is a normal graph with△(G)≥5 and mad(G) 〈 3-2/(△(G)), then χ'Σ(G)≤△(G) + 1. This improves the previous results and the bound △(G) + 1 is sharp.
基金Supported by the National Natural Science Foundation of China(No.10771091No.61163010)Ningxia University Science Research Foundation(No.(E)ndzr09-15)
文摘Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χ áve (G) of some special graphs and present a conjecture.
基金Supported by the National Natural Science Foundation of China(No.10771091)
文摘The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χt(G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one', then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χet(G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order.
基金supported by National Natural Science Foundation of China(Grant No.11901318)the Fundamental Research Funds for the Central Universities,Nankai University(Grant No.63191425)supported by National Natural Science Foundation of China(Grant Nos.11571149 and 11971205)
文摘A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3.The star chromatic indexχ'_(st)(G)of a graph G is the smallest integer k such that G has a star k-edge-coloring.The list star chromatic index ch'st(G)is defined analogously.The star edge coloring problem is known to be NP-complete,and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear.In this paper,we study,in contrast,the best linear upper bound for sparse graph classes.We show that for everyε>0 there exists a constant c(ε)such that if mad(G)<8/3-ε,then■and the coefficient 3/2 ofΔis the best possible.The proof applies a newly developed coloring extension method by assigning color sets with different sizes.