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Incidence Coloring of Outer-1-planar graphs

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Acta Mathematicae Applicatae Sinica 2024年第3期40卷 840-848页

作者： Meng-ke QI Xin ZHANG School of Mathematics and Statistics Xidian UniversityXi'an 710071China

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.

关键词： incidence coloring outer--planar graph planar graph

planar graphs whose Second Largest Eigenvalue Smaller than ■

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Journal of Mathematical Research with Applications 2024年第1期44卷 1-6页

作者： Zihao GENG Jiming GUO Department of Mathematics East China University of Science and Technology

A graph G is said to be planar if G can be drawn on the plane in such a way that its edges intersect only at their endpoints.In this paper,all the planar graphs without isolated vertices whose second largest eigenvalu... 详细信息

关键词： planar graph the second largest eigenvalue complete product

On the Enhanced Power graph of a Semigroup

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Algebra Colloquium 2024年第1期31卷 83-96页

作者： Sandeep Dalal Jitender Kumart Siddharth Singh Department of Mathematics Birla Institute of Technology and Science Pilani PilaniIndia

The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.

关键词： monogenic semigroup completely regular semigroup planar graph enhanced powergraph

SOME APPLICATIONS OF planar graph IN KNOT THEORY

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Acta Mathematica Scientia 2012年第2期32卷 663-671页

作者：程志云高红铸 School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education

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 .

关键词： planar graph almost planar Seifert surface amphicheiral knot alternating knot

Equitable Cluster Partition of planar graphs with Girth at Least 12

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Journal of Mathematical Research with Applications 2024年第2期44卷 152-160页

作者： Xiaoling LIU Lei SUN Wei ZHENG Shandong Normal University Department of Mathematics and Statistics

An equitable(Ok1, Ok2,..., Okm)-partition of a graph G, which is also called a k cluster m-partition, is the partition of V(G) into m non-empty subsets V1, V2,..., Vm such that for every integer i in {1, 2,..., m}, G[Vi] is a graph with components of order at most k, and for each distinct pair i, j in {1,..., m}, there is-1 ≤ |Vi|-|Vj| ≤ 1. In this paper, we proved that every planar graph G with minimum degree δ(G) ≥ 2 and girth g(G) ≥ 12 admits an equitable(O71, O72,..., O7m)-partition, for any integer m ≥ 2.

关键词： equitable cluster partition planar graph girth discharging

Extremal P_(8)-Free/P_(9)-Free planar graphs

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Journal of the Operations Research Society of China 2023年第3期11卷 451-457页

作者： Yong-Xin Lan Yong-Tang Shi School of Science Hebei University of TechnologyTianjin300401China Center for Combinatorics and LPMC Nankai UniversityTianjin300071China

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.

关键词： Turán number Extremal graph planar graph

Multi-scale visual analysis of cycle characteristics in spatially-embedded graphs

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Visual Informatics 2023年第3期7卷 49-58页

作者： Farhan Rasheed Talha Bin Masood Tejas G.Murthy Vijay Natarajan Ingrid Hotz Department of Science and Technology LinkOing UniversityBredgatan 33NorrkOing60221Sweden Department of Civil Engineering Indian Institute of ScienceBangalore560012India Department of Computer Science and Automation Indian Institute of ScienceBangalore560012India

We present a visual analysis environment based on a multi-scale partitioning of a 2d domain intoregions bounded by cycles in weighted planar embedded graphs.The work has been inspired by anapplication in granular materials research,where the question of scale plays a fundamental role inthe analysis of material properties.We propose an efficient algorithm to extract the hierarchical cyclestructure using persistent homology.The core of the algorithm is a filtration on a dual graph exploitingAlexander’s duality.The resulting partitioning is the basis for the derivation of statistical properties thatcan be explored in a visual environment.We demonstrate the proposed pipeline on a few syntheticand one real-world dataset.

关键词： Visual data analysis planar graph Force network Granular materials Persistence homology Force loops Computational geometry

List Injective Coloring of planar graphs

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Acta Mathematicae Applicatae Sinica 2022年第3期38卷 614-626页

作者： Yin-dong JIN Lian-ying MIAO School of Mathematics China University of Mining and TechnologyXuzhou 221116China

A k-coloring of a graph G is a mapping c:V(G)→{1,2,···,k}.The coloring c is called injective if any two vertices have a common neighbor get distinct colors.A graph G is injectively k-choosable if for any color list L of admissible colors on V(G)of size k allows an injective coloringφsuch thatφ(v)∈L(v)for each v∈V(G).Letχ;(G),χ;(G)denote the injective chromatic number and injective choosability number of G,respectively.In this paper,we show thatχ;(G)≤△+4 if△≥22 andχ;(G)≤△+5 if△≥15,where G is a triangle-free planar graph and without intersecting 4-cycles.

关键词： planar graph injective coloring list coloring

Injective coloring of planar graphs with girth 5

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Frontiers of Mathematics in China 2022年第3期17卷 473-484页

作者： Yuehua BU Piaopiao YE School of Mathematical Sciences Zhejiang Normal UniversityJinhua 321004China Xingzhi College Zhejiang Normal UniversityJinhua 321004China

A coloring of a graph G is injective if its restriction to the neighbour of any vertex is injective.The injective chromatic number x_(i)(G)of a graph G is the least k such that there is an injective k-coloring.In this... 详细信息

关键词： planar graph girth injective coloring face