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On the (Δ + 2)-Total-Colorability of Planar Graphs with 7-Cycles Containing at Most Two Chords
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作者 Jian Chang Jingru Liu Fan Zhang 《Journal of Applied Mathematics and Physics》 2024年第7期2702-2710,共9页
The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove tha... The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords. 展开更多
关键词 planar graph 7-Cycle 8-Totally-Colorable Maximum Degree
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Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making 被引量:1
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作者 P.Chellamani D.Ajay +1 位作者 Mohammed M.Al-Shamiri Rashad Ismail 《Computers, Materials & Continua》 SCIE EI 2023年第6期4935-4953,共19页
Graph theory has a significant impact and is crucial in the structure of many real-life situations.To simulate uncertainty and ambiguity,many extensions of graph theoretical notions were created.Planar graphs play a v... Graph theory has a significant impact and is crucial in the structure of many real-life situations.To simulate uncertainty and ambiguity,many extensions of graph theoretical notions were created.Planar graphs play a vital role in modelling which has the property of non-crossing edges.Although crossing edges benefit,they have some drawbacks,which paved the way for the introduction of planar graphs.The overall purpose of the study is to contribute to the conceptual development of the Pythagorean Neutrosophic graph.The basic methodology of our research is the incorporation of the analogous concepts of planar graphs in the Pythagorean Neutrosophic graphs.The significant finding of our research is the introduction of Pythagorean Neutrosophic Planar graphs,a conceptual blending of Pythagorean Neutro-sophic and Planar graphs.The idea of Pythagorean Neutrosophic multigraphs and dual graphs are also introduced to deal with the ambiguous situations.This paper investigates the Pythagorean Neutrosophic planar values,which form the edges of the Pythagorean neutrosophic graphs.The concept of Pythagorean Neutrosophic dual graphs,isomorphism,co-weak and weak isomorphism have also been explored for Pythagorean Neutrosophic planar graphs.A decision-making algorithm was proposed with a numerical illustra-tion by using the Pythagorean Neutrosophic fuzzy graph. 展开更多
关键词 Pythagorean neutrosophic planar graph planarity value ISOMORPHISM dual graphs MULTIgraph
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Power domination in planar graphs with small diameter 被引量:1
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作者 赵敏 康丽英 《Journal of Shanghai University(English Edition)》 CAS 2007年第3期218-222,共5页
The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In t... The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In this paper, it was shown that the power domination number of an outerplanar graph with the diameter two or a 2-connected outerplanar graph with the diameter three is precisely one. Upper bounds on the power domination number for a general planar graph with the diameter two or three were determined as an immediate consequences of results proven by Dorfling, et al. Also, an infinite family of outerplanar graphs with the diameter four having arbitrarily large power domination numbers were given. 展开更多
关键词 graph power domination planar graph outerplanar graph
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SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY
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作者 程志云 高红铸 《Acta Mathematica Scientia》 SCIE CSCD 2012年第2期663-671,共9页
The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship betwe... The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n 〉 4. This gives an affirmative answer to problem 1.66(B) on Kirby's problem list . 展开更多
关键词 planar graph almost planar Seifert surface amphicheiral knot alternating knot
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A NOTE ON STRONG EMBEDDINGS OF MAXIMAL PLANAR GRAPHS ON NON ORIENTABLE SURFACES
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作者 Liu Tongyin Liu Yanpei Dept.ofMath.,NorthernJiaotongUniv.,Beijing100044. 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2001年第2期111-114,共4页
In this paper, it is shown that for every maximal planar graph G=(V,E) , a strong embedding on some non orientable surface with genus at most |V(G)|-22 is admitted such that the surface dual of G is also a... In this paper, it is shown that for every maximal planar graph G=(V,E) , a strong embedding on some non orientable surface with genus at most |V(G)|-22 is admitted such that the surface dual of G is also a planar graph. As a corollary, an interpolation theorem for strong embeddings of G on non orientable surfaces is obtained. 展开更多
关键词 SURFACE strong embedding maximal planar graph.
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Improper Choosability of Planar Graphs without 6-circuits
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作者 ZHANG Hai-hui 《Chinese Quarterly Journal of Mathematics》 CSCD 2010年第4期510-514,共5页
A graph G is called(k,d)*-choosable if for every list assignment L satisfying |L(v)|=k for all v ∈ V(G),there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same co... A graph G is called(k,d)*-choosable if for every list assignment L satisfying |L(v)|=k for all v ∈ V(G),there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself.In this paper,it is shown that every planar graph without 6-circuits and a triangle adjacent to itself or a quadrangle is(3,1)*-choosable. 展开更多
关键词 TRIANGLE CIRCUIT improper choosability planar graph
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Bondage Number of 1-Planar Graph 被引量:1
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作者 Qiaoling Ma Sumei Zhang Jihui Wang 《Applied Mathematics》 2010年第2期101-103,共3页
The bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G. In this paper, we prove that ... The bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G. In this paper, we prove that for a 1-planar graph G. 展开更多
关键词 DOMINATION NUMBER Bondage NUMBER 1-planar graph Combinatorial PROBLEM
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Incidence Coloring of Outer-1-planar Graphs
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作者 Meng-ke QI Xin ZHANG 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2024年第3期840-848,共9页
A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge.It is known that every outer-1-planar graph is a planar partial3-tree... A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge.It is known that every outer-1-planar graph is a planar partial3-tree.In this paper,we conjecture that every planar graph G has a proper incidence(Δ(G)+2)-coloring and confirm it for outer-1-planar graphs with maximum degree at least 8 or with girth at least 4.Specifically,we prove that every outer-1-planar graph G has an incidence(Δ(G)+3,2)-coloring,and every outer-1-planar graph G with maximum degree at least 8 or with girth at least 4 has an incidence(Δ(G)+2,2)-coloring. 展开更多
关键词 incidence coloring outer--planar graph planar graph
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Edge Colorings of Planar Graphs without 6-Cycles with Two Chords
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作者 Ling Xue Jianliang Wu 《Open Journal of Discrete Mathematics》 2013年第2期83-85,共3页
It is proved here that if a planar graph has maximum degree at least 6 and any 6-cycle contains at most one chord, then it is of class 1.
关键词 EDGE COLORING planar graph CYCLE Class 1
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Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10
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作者 Dong-han Zhang You Lu +1 位作者 Sheng-gui Zhang Li Zhang 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2024年第1期211-224,共14页
A neighbor sum distinguishing(NSD)total coloringφof G is a proper total coloring of G such thatΣz∈EG(u)U{u}φ(z)≠Σz∈EG(v)U{v}φ(z)for each edge uv∈E(G),where EG(u)is the set of edges incident with a vertex u.In... A neighbor sum distinguishing(NSD)total coloringφof G is a proper total coloring of G such thatΣz∈EG(u)U{u}φ(z)≠Σz∈EG(v)U{v}φ(z)for each edge uv∈E(G),where EG(u)is the set of edges incident with a vertex u.In 2015,Pilśniak and Wozniak conjectured that every graph with maximum degreeΔhas an NSD total(Δ+3)-coloring.Recently,Yang et al.proved that the conjecture holds for planar graphs withΔ≥10,and Qu et al.proved that the list version of the conjecture also holds for planar graphs withΔ≥13.In this paper,we improve their results and prove that the list version of the conjecture holds for planar graphs withΔ≥10. 展开更多
关键词 planar graphs neighbor sum distinguishing total choosibility combinatorial nullstellensatz discharging method
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Fractional Coloring Planar Graphs under Steinberg-type Conditions
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作者 Xiao Lan HU Jia Ao LI 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2023年第5期904-922,共19页
A Steinberg-type conjecture on circular coloring is recently proposed that for any prime p≥5,every planar graph of girth p without cycles of length from p+1 to p(p-2)is Cp-colorable(that is,it admits a homomorphism t... A Steinberg-type conjecture on circular coloring is recently proposed that for any prime p≥5,every planar graph of girth p without cycles of length from p+1 to p(p-2)is Cp-colorable(that is,it admits a homomorphism to the odd cycle Cp).The assumption of p≥5 being prime number is necessary,and this conjecture implies a special case of Jaeger’s Conjecture that every planar graph of girth 2p-2 is Cp-colorable for prime p≥5.In this paper,combining our previous results,we show the fractional coloring version of this conjecture is true.Particularly,the p=5 case of our fractional coloring result shows that every planar graph of girth 5 without cycles of length from 6 to 15 admits a homomorphism to the Petersen graph. 展开更多
关键词 Fractional coloring circular coloring planar graphs GIRTH HOMOMORPHISM
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Injective △+2 Coloring of Planar Graph Without Short Cycles
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作者 Ying CHEN Lan TAO Li ZHANG 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2023年第4期1009-1031,共23页
A coloring of graph G is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective ch... A coloring of graph G is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number χi(G) of G is the least integer k such that G has an injective k-coloring. In this paper, we prove that(1) if G is a planar graph with girth g ≥ 6 and maximum degree △ ≥ 7, then χi(G) ≤ △ + 2;(2) if G is a planar graph with △ ≥ 24 and without 3,4,7-cycles, then χi(G) ≤ △ + 2. 展开更多
关键词 injective coloring planar graph maximum degree CYCLE
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Extremal P_(8)-Free/P_(9)-Free Planar Graphs
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作者 Yong-Xin Lan Yong-Tang Shi 《Journal of the Operations Research Society of China》 EI CSCD 2023年第3期451-457,共7页
An H-free graph is a graph not containing the given graph H as a subgraph.It is well known that the Turán number ex(n,H)is the maximum number of edges in an H-free graph on n vertices.Based on this definition,we ... An H-free graph is a graph not containing the given graph H as a subgraph.It is well known that the Turán number ex(n,H)is the maximum number of edges in an H-free graph on n vertices.Based on this definition,we define ex_(P)(n,H)to restrict the graph classes to planar graphs,that is,ex_(P)(n,H)=max{|E(G)|:G∈G,where G is a family of all H-free planar graphs on n vertices.Obviously,we have ex_(P)(n,H)=3n−6 if the graph H is not a planar graph.The study is initiated by Dowden(J Graph Theory 83:213–230,2016),who obtained some results when H is considered as C_(4)or C_(5).In this paper,we determine the values of ex_(P)(n,Pk)with k∈{8,9},where Pk is a path with k vertices. 展开更多
关键词 Turán number Extremal graph planar graph
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On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number 被引量:1
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作者 Xiao Ming PI 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2018年第5期911-920,共10页
Let G = (V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function (SCDF) of G if ∑e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination numbe... Let G = (V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function (SCDF) of G if ∑e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc(G) = min{∑e∈E f(e)| f is an SCDF of G}. This paper will characterize all maxima] planar graphs G with order n ≥ 6 and γ′sc(G) =n. 展开更多
关键词 Domination number signed cycle domination function signed cycle domination number planar graph maximal planar graph
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Some Planar Graphs with Star Chromatic Number Between Three and Four
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作者 李德明 《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2001年第4期500-504,共5页
We construct sonic infinite family of planar graphs with star chromatic number, where, partially answering a question of Vince,
关键词 (k d)-coloring star chromatic number planar graph
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Spectral Radius of Hamiltonian Planar Graphs and Outerplanar Graphs
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作者 周建 林翠琴 胡冠章 《Tsinghua Science and Technology》 SCIE EI CAS 2001年第4期350-354,共5页
The spectral radius is an important parameter of a graph related to networks. A method for estimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of a Hamiltonian planar g... The spectral radius is an important parameter of a graph related to networks. A method for estimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of a Hamiltonian planar graph of order n≥4 is less than or equal to 2+3n-11 and the spectral radius of the outerplanar graph of order n≥6 is less than or equal to 22+n-5, which are improvements over previous results. A direction for further study is then suggested. 展开更多
关键词 spectral radius Hamiltonian planar graphs maximal outerplanar graphs
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Acyclic edge colorings of planar graphs and series-parallel graphs 被引量:24
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作者 HOU JianFeng WU JianLiang +1 位作者 LIU GuiZhen LIU Bin 《Science China Mathematics》 SCIE 2009年第3期605-616,共12页
A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a (G), is the least number of colors in an acyclic edge coloring of G. Alon... A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a (G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a (G) Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a (G) max{2Δ(G) + 2, Δ(G) + 22} if g(G) 3, a (G) Δ(G) + 2 if g(G) 5, a (G) Δ(G) + 1 if g(G) 7, and a (G) = Δ(G) if g(G) 16 and Δ(G) 3. For series-parallel graphs G, we have a (G) Δ(G) + 1. 展开更多
关键词 acyclic coloring planar graph GIRTH series-parallel graph 05C15
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Total colorings of planar graphs with maximum degree at least 8 被引量:6
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作者 SHEN Lan WANG YingQian 《Science China Mathematics》 SCIE 2009年第8期1733-1742,共10页
Planar graphs with maximum degree Δ 8 and without 5- or 6-cycles with chords are proved to be (Δ + 1)-totally-colorable.
关键词 planar graph total coloring maximum degree CYCLE CHORD 05C15 68R10
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On total chromatic number of planar graphs without 4-cycles 被引量:7
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作者 Min-le SHANGGUAN 《Science China Mathematics》 SCIE 2007年第1期81-86,共6页
Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has... Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs is Δ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then x ve (G) ? 8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture. 展开更多
关键词 total chromatic number planar graph F 5-subgraph 05C40
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Planar graphs with maximum degree 8 and without intersecting chordal 4-cycles are 9-totally colorable 被引量:5
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作者 CAI JianSheng WANG GuangHui YAN GuiYing 《Science China Mathematics》 SCIE 2012年第12期2601-2612,共12页
The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum deg... The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9. 展开更多
关键词 total coloring planar graph chordal 4-cycles TRIANGLES
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