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Group Connectivity and Group Colorings of Graphs -- A Survey

Acta Mathematica Sinica,English Series 2011年第3期27卷 405-434页

作者： hong-jian lai Xiangwen LI Yehong SHAO Mingquan ZHAN College of Mathematics and System Sciences Xinjiang University Urumqi 830046 P. R. China Department of Mathematics West Virginia University Morgantown WV26506 USA Department of Mathematics Huazhong Normal University Wuhan 430070 P. R. China Department of Mathematics Ohio University Southern Campus Ironton OH 45638 USA Department of Mathematics Millersville University of Pennsylvania Millersville PA 17551 USA

In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an （additive） abelian group and A＊ = A - {0}. The graph G is A-connected if G has an orientation D（G） such that for every map b ： V（G） → A satisfying ∑v∈V（G）b（v）： 0, there is a function f ： E（G） → A＊ such that for each vertex v ∈ V（G）, the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v is equal to b（v）. The group coloring of a graph arises from the dual concept of group connectivity. There have been lots of investigations on these subjects. This survey provides a summary of researches on group connectivity and group colorings of graphs. It contains the following sections. 1. Nowhere-zero Flows and Group Connectivity of Graphs 2. Complete Families and A-reductions 3. Reductions with Edge-deletions, Vertex-deletions and Vertex-splitting 4. Group Colorings as a Dual Concept of Group Connectivity 5. Brooks Theorem, Its Variations and Dual Forms 6. Planar Graphs 7. Group Connectivity of Graphs 7.1 Highly Connected Graphs and Collapsible Graphs 7.2 Degrees Conditions 7.3 Complementary Graphs 7.4 Products of Graphs 7.5 Graphs with Diameter at Most 2 7.6 Line Graphs and Claw-Free Graphs 7.7 Triangular Graphs 7.8 Claw-decompositions and All Tutte-orientations

关键词： Group connectivity group connectivity number group coloring group chromatic number

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Supereulerian Graphs and the Petersen Graph

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Acta Mathematica Sinica,English Series 2014年第2期30卷 291-304页

作者： Xiao Min LI Lan LEI hong-jian lai Meng ZHANG Department of Mathematics and Statistics Chongqing Technology and Business University Department of Mathematics West Virginia University College of Mathematics and System Sciences Xinjiang University

A graphG is supereulerian if G has a spanning eulerian *** et al.[*** Theory,1,79–84（1977）]proposed the problem of characterizing supereulerian *** this paper,we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen *** extends a former result of Catlin and lai[***,Ser.B,66,123–139（1996）].

关键词： Supereulerian graphs petersen graph edge-cut reduction contraction

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On the Sizes of k-edge-maximal r-uniform Hypergraphs

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Acta Mathematicae Applicatae Sinica 2022年第3期38卷 532-539页

作者： Ying-zhi TIAN hong-jian lai Ji-xiang MENG Li-qiong XU College of Mathematics and System Sciences Xinjiang UniversityUrumqi 830046China Department of Mathematics West Virginia UniversityMorgantownWV 26506USA School of Science Jimei UniversityXiamen 361021China

Let H=(V,E)be a hypergraph,where V is a set of vertices and E is a set of non-empty subsets of V called *** all edges of H have the same cardinality r,then H is an r-uniform hypergraph;if E consists of all r-subsets of V,then H is a complete r-uniform hypergraph,denoted by K_(n)^(r),where n=|V|.A hypergraph H′=(V′,E′)is called a subhypergraph of H=(V,E)if V′⊆V and E′⊆*** edge-connectivity of a hypergraph H is the cardinality of a minimum edge set F⊆E such that H−F is not connected,where H−F=(V,E\F).An r-uniform hypergraph H=(V,E)is k-edge-maximal if every subhypergraph of H has edge-connectivity at most k,but for any edge e∈E(K_(n)^(r))\E(H),H+e contains at least one subhypergraph with edge-connectivity at least k+*** k and r be integers with k≥2 and r≥2,and let t=t(k,r)be the largest integer such that(t−1 r−1)≤*** is,t is the integer satisfying(t−1 r−1)≤k<(t r−1).We prove that if H is an r-uniform k-edge-maximal hypergraph such that n=|V(H)|≥t,then(i)|E(H)|≤(t r)+(n−t)k,and this bound is best possible;(ii)|E(H)|≥(n−1)k−((t−1)k−(t r))[n/t],and this bound is best possible.

关键词： Edge-connectivity k-edge-maximal hypergraphs r-uniform hypergraphs

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Multigraphic Degree Sequences and Hamiltonian-connected Line Graphs

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Acta Mathematicae Applicatae Sinica 2021年第4期37卷 800-806页

作者： Ren-sen MA Ai-mei YU Ke-ke WANG hong-jian lai Department of Mathematics Beijing Jiaotong UniversityBeijing100044China Department of Mathematics Embry-Riddle Aeronautical UniversityPrescottAZUSA Department of Mathematics West Virginia UniversityMorgantownWVUSA

Let G be a *** that e=u1v1 and e′=u2v2 are two edges of *** e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-*** design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is *** this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥*** this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.

关键词： strongly spanning trailable graphs multigraphic degree sequence hamiltonian-connected graphs line graph

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Reinforcing a Matroid to Have k Disjoint Bases

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Applied Mathematics 2010年第3期1卷 244-249页

作者： hong-jian lai Ping Li Yanting Liang Jinquan Xu 不详

Let denote the maximum number of disjoint bases in a matroid . For a connected graph , let , where is the cycle matroid of . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with . Edmonds generalizes this theorem to matroids. In [1] and [2], for a matroid with , elements with the property that have been characterized in terms of matroid invariants such as strength and -partitions. In this paper, we consider matroids with , and determine the minimum of , where is a matroid that contains as a restriction with both and . This minimum is expressed as a function of certain invariants of , as well as a min-max formula. These are applied to imply former results of Haas [3] and of Liu et al. [4].

关键词： Disjoint Bases Edge-Disjoint Spanning Trees Spanning Tree Packing Numbers Strength

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On a Class of Supereulerian Digraphs

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Applied Mathematics 2016年第3期7卷 320-326页

作者： Khalid A. Alsatami Xindong Zhang Juan Liu hong-jian lai Department of Mathematics College of Science Qassim University Buraydah KSA College of Mathematics Sciences Xinjiang Normal University Urumqi China Department of Mathematics West Virginia University Morgantown WV USA

The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.

关键词： Supereulerian Digraph 2-Sums Arc-Strong-Connectivity Hamiltonian-Connected Digraphs

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α_(2-)独立数为2的有向图中的迹,路和圈

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数学学报（中文版） 2024年第1期67卷 137-150页

作者：张新东杨洪赖虹建刘娟贵州财经大学大数据统计学院贵阳550025 新疆大学数学与系统科学学院乌鲁木齐830046 西弗吉尼亚大学数学系摩根敦WV26506

设α_(2-)(D)=max{|X|:X■V(D)且D[X]不含有向2-圈}是有向图D的α_(2-)(D)-独立数.在文献[*** ***.,42(1981)231-251]中,Thomassen构造了满足κ(D)=α(D)的非哈密尔顿有向图D,以此证明Chvátal-Erdös定理在有向图情形下不能得到自然推广.Bang-Jensen和Thomassé提出如下猜想:每一个满足弧强连通度大于等于其独立数的有向图一定包含生成闭迹.对于满足弧强连通度大于等于其α_(2-)(D)-独立数的有向图是否包含生成迹这一问题,目前仍未解决.如果对于D中的任意两个顶点x和y,D包含生成(x,y)-迹,或者生成(y,x)-迹,则称有向图D是弱迹连通的.如果对于D中的任意两个顶点x和y,D既包含生成(x,y)-迹又包含生成(y,x)-迹,则称D是强迹连通的.本文在确定两个强连通有向图类M和H的基础上,研究了在满足α_(2-)(D)=2条件下,有向图D的相关结果,并得到以下结论:(ⅰ)D是哈密尔顿的当且仅当D■M.(ⅱ)D是弱迹连通的.(ⅲ)D是强迹连通的当且仅当D?H.特别地,每一个满足α_(2-)(D)=2的强连通有向图D包含哈密尔顿路,并且每一个满足α_(2-)(D)=2的2-强连通有向图D是强迹连通的.

关键词： α_(2)(D)-独立集哈密尔顿圈弱迹连通强迹连通

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基于计划行为理论大学生环境友好行为意向研究

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内蒙古财经大学学报 2017年第2期15卷 59-64页

作者：洪坚仁赖美娇林珊福州外语外贸学院工程系福建福州350202

环境保护在世界各地透过各种管道已推广一段时间,然而仍有不少地方对于环境保护充满好奇及怀疑,大多停留于宣传口号上。本研究以计划行为理论为架构,探讨大学生环境保护态度、环境保护主观规范、环境保护知觉行为控制对环境友好行为意... 详细信息

环境保护在世界各地透过各种管道已推广一段时间,然而仍有不少地方对于环境保护充满好奇及怀疑,大多停留于宣传口号上。本研究以计划行为理论为架构,探讨大学生环境保护态度、环境保护主观规范、环境保护知觉行为控制对环境友好行为意向的影响。结果发现大学生行为态度、行为知觉、行为控制对环境友好行为意向有显著影响,而大学生环境保护主观规范对环境友好行为意向不具显著的影响,分析其原因可能是大学生处于叛逆青年期,自我意识高,有自己的主见。

关键词：环境保护计划行为理论环境友好行为意向

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Preparation and Molecular Structure of (SCZ)(TNPG)·2H_2O

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Chinese Journal of Structural Chemistry 2005年第8期24卷 973-979页

作者： CHEN hong-Yan ZHANG Tong-lai ZHANG jian-Guo State Key Laboratory of Prevention and Control of Explosion Disasters and School of Science Beijing Institute of Technology Beijing 100081 China

The （SCZ）（TNPG）.2H2O prepared by mixing semicarbazide （SCZ） and trinitrophloroglucinol （TNPG） was characterized by elemental analysis and IR measurement, and its crystal structure was determined by X-ray single-crystal diffraction analysis. The crystal belongs to triclinic,space group P1, （NH2CONHNH3）^＋（C6H2N3O9）^-·2H2O, Mr = 372.23, a = 0.68853（14）, b =0.93966（19）, c = 1.1925（2） nm, α = 67.48（3）, β = 77.56（3）, γ = 78.93（3）°, V = 0.6908（2） nm^3, Z = 2,Dc = 1.789 g/cm^3, Mr = 373.23, F（000） = 384, S = 0.999 and μ（MoKa） = 0.172 mm^-1. The final R and wR are 0.0394 and 0.1057 for 1724 observed reflections with I 〉 2σ（I）. It is concluded that （SCZ）（TNPG） 2H2O is an ionic compound composed of a cation SCZ^＋, an anion TNPG- and two water molecules. The TNPG anion and SCZ^＋ cation are bonded together by electrostatic attraction and hydrogen bonds, and the compound structure is stable. The thermal decomposition of （SCZ）（TNPG） 2H2O was studied by using TG-DTG and DSC techniques with a heating rate of 10 ℃/min, showing the compound contains one endothermic process of dehydrating stage and two intensive exothermic decomposition stages. The enthalpy of exothermic decomposition reaction is 452.31 kJ/mol.

关键词： semicarbazide trinitrophloroglucinol preparation crystal structure

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Fouling-release Property of Water-filled Porous Elastomers

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Chinese Journal of Chemical Physics 2012年第3期25卷 330-334,374页

作者： lai-yong Xie Fei hong Chuan-xin He jian-hong Liu Chi Wu Hefei National Laboratory for Physical Sciences at Microscale Department of Chemical PhysicsUniversity of Science and Technology of China Hefei 230025 China School of Chemistry and Chemical Engineering Shenzhen University Shenzhen 518060 China Department of Chemistry The Chinese University of Hong Kong Hong Kong

Since the fouling-releasing ability of silicone elastomers increased as their modulus decreases, we designed and prepared composites with embedded tiny NaC1 crytals that were soluble after their immersion in water, resulting in water-filled porous elastomers. The scanning electron microscope images confirmed such a designed water-filling porous structure. The existence of many micro-drops of water in these specially designed elastomers decreased the shear storage modulus and increased the loss factors. The decrease of shear modulus plays a leading role here and is directly related to a lower critical peeling-off stress of a pseudo-barnacle on them. Therefore, such a novel preparation with cheap salts instead of an expensive silicone provides a better way to make fouling-release paints with a lower modulus, a lower critical peeling-off stress and a better fouling-release property without a significant decrease of the cross-linking density.

关键词： Fouling-release Porous silicone elastomer Soluble salt Pseudo-barnacle ad-hesion NaCl Micro-drop of water

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