边染色临界图独立数的新下界

1968年;Vizing提出猜想:如果图G是Delta-临界图, 则其独立数alpha (G)满足alpha (G)leqslantdfrac{n}{2}。这一猜想至今仍未解决。本文对于不含度点的最大度较小的临界图, 证明当最大度Deltain{3, 4, 5, 6}时, 独立数alpha (G)leqslantdfrac{7Delta-6}{12Delta-6}V;当Deltain{7, 8, 9}时, 独立数alpha (G)leqslantdfrac{4Delta-3}{7Delta-3}V。     

几类积图的团染色数

设G=（VE）为简单图,V和E分别表示图的点集和边集．图G的一个k-团染色是指点集V到色集{1,2,…,k）的一个映射,使得G的每个至少含两个点的极大团都至少有两种颜色．分别给出了任意两个图的团色数与它们通过笛卡尔积、Kronecker积、强直积或字典积运算后得到的积图的团色数之间的关系．     

几类二部图的pebbling数

Chung定义了图G上的一个pebbling移动是从一个顶点移走两个pebble而把其中的一个移到与其相邻的一个顶点上.连通图G的pebbling数f(G)是最小的正整数n,使得不管n个pebble如何放置在G的顶点上,总可以通过一系列的pebbling移动把一个pebble移到G的任意一个顶点上.Graham猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H).作者们验证了三类二部图的2-pebbling性质以及当H为此类二部图,G为一个2-pebbling性质的图时,Graham猜想成立.     

几类Mycielske图的Smarandchely邻点可区别染色

本文根据路和圈、星的Mycielski图的结构性质.利用穷染递推,反证的方法,研究了图M(Pm)和M(Cm),以及M(Sm)的Smarandchely-邻点可区别边染色,得到了相应的边色数,分别给出它们的一种染色方案,推广了文献[9]的结果.     

赋权图的圆染色与区间染色

赋权图的区间染色的定义与赋权图的圆染色的定义非常类型,唯一的区别就是将Ｇ的顶点对应圆周上的孤换为Ｇ的顶点对应区间上的子区间,讨论了赋权的圆染色与区染色的关系。     

若干Mycielski图的均匀染色

如果图G的一个正常顶点染色满足任两个色类中的顶点数相差不超过1,则称为G的均匀染色.研究了一些Mycielski图的均匀染色,给出了路、圈、完全图和广义星图的Mycielski图的均匀色数.     

一些特殊图的Mycielski图的点可区别均匀全色数

讨论了路、圈、星的Mycielski图的点可区别均匀全染色问题,得到了其点可区别均匀全色数.     

一些联图的均匀全染色

对-个正常的全染色满足各种颜色所染元素数(点或边)相差不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数.本文证明了图在若干情况下的均匀全色数定理,得到了Cm VSn,CmVFn和CmVWn的均匀全色数.     

稀疏图与系列平行图的列表动态染色

图的(列表)动态染色模型可用于解决信道分配中的一些关键问题,是图论和理论计算机科学领域的一个重要的研究方向.Kim和Park (2011)给出了任何最大平均度小于8/3的图的列表动态色数至多为4的证明.然而,由于具有5个顶点的圈C₅的最大平均度为2且列表动态色数为5,因此Kim和Park的上述结论是错误的.基于此,本文证明了任何最大平均度小于8/3的普通图(每个连通分支都不与C₅同构的图)的列表动态色数至多为4,且该上界4是最优的,从而对Kim和Park的结果进行了修正.与此同时,本文证明了如果图G是系列平行图,则当其是普通图时,其列表动态色数至多为4,且该上界4是最优的,当其不是普通图时,其列表动态色数恰好为5,从而将Song等人(2014)的结果“任何系列平行图的列表动态色数至多为6”进行了改进.     

一些联图的均匀全染色

对一个正常的全染色满足各种颜色所染元素数(点或边)相差不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数.本文证明了图在若干情况下的均匀全色数定理,得到C_m∨S_nC_m∨ F_n和C∨W_n的均匀全色数.     

On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.     

若干图的广义Mycielski图的边色数

设图G(V,E)为简单图,V(Mn(G))={v01,v02,…,v0p;v11,v12,…,v1p;…,vn1,vn2,…,vnp}EMn(G))=E(G)∪vijv(i+1)kv0 jv0k∈E(G),1 j,k p,i=0,1,…,n-1称Mn(G)为G的n串广义M ycielsk i图,其中n为自然数,V(G)={v01,v02,…,v0p}.本文得到了路、圈、扇、轮、星图的广义M ycielsk i图的边色数.     

两类Mycielski图及一些平面图的均匀全色数

对于图G(V,E)的正常k-全染色φ称为G(V,E)的k-均匀全染色,当且仅当任意两个色类中的元素总数至多相差1.xvee(G)=m in{k存在图G的一个k-均匀全染色}称为G的均匀全色数.本文得到了两类M ycielsk i图及圈,轮图和扇形的均匀全色数.     

On b-perfect Chordal Graphs

The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbour in all other color classes. We characterize the class of chordal graphs for which the b-chromatic number is equal to the chromatic number for every induced subgraph. This research was supported by Algerian-French program CMEP/Tassili 05 MDU 639.     

The Basis Number of Some Special Non-Planar Graphs

The basis number of a graph G was defined by Schmeichel to be the least integer h such that G has an h-fold basis for its cycle space. He proved that for m, n 5, the basis number b(K _m,n ) of the complete bipartite graph K _m,n is equal to 4 except for K _6,10, K _5,n and K _6,n with n = 5, 6, 7, 8. We determine the basis number of some particular non-planar graphs such as K _5,n and K _6,n , n = 5, 6, 7, 8, and r-cages for r = 5, 6, 7, 8, and the Robertson graph.     

English Decycling Number of Some Graphs

1.IntroductionInthispaper,weonlydiscusssimplegraph(withneithermulti-edgenorloop).TheterminologiesnotexplainedcanbeseeninII].Thecyclerankofagraphistheminimumnumberofedgesthatmustberemovedinordertoeliminateallofthecyclesinthegraph.IfGhaspvenices,qedges...     

若干星的冠图的邻点可区别V-全染色

根据星与圈(星、扇、轮、路)构造的冠图的结构性质,应用分析和构造函数法研究了邻点可区别V-全染色,得到了S_n·C_m,S_n·S_m,S_n·F_m,S_n·W_m,S_n·P_m的邻点可区别V-全色数.     

The b-Chromatic Number of Cubic Graphs

The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex adjacent to some vertex in all the other color classes. It is proved that with four exceptions, the b-chromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K _3,3, the prism over K ₃, and one more sporadic example on 10 vertices.     

b-Chromatic Number of Cartesian Product of Some Families of Graphs

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. It is known that for any two graphs G and H, ${b(G \square H) \geq {\rm {max}} \{b(G), b(H)\}}$ , where ${\square}$ stands for the Cartesian product. In this paper, we determine some families of graphs G and H for which strict inequality holds. More precisely, we show that for these graphs G and H, ${b(G \square H) \geq b(G) + b(H) - 1}$ . This generalizes one of the results due to Kouider and Mahéo.     

On Domination Number of 4-Regular Graphs

Let G be a simple graph. A subset S V is a dominating set of G, if for any vertex v V – S there exists a vertex u S such that uv E(G). The domination number, denoted by (G), is the minimum cardinality of a dominating set. In this paper we prove that if G is a 4-regular graph with order n, then (G) ⁴/₁₁ n