关于图的符号边全控制数

Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination number γ st （G） of G is defined as γ st （G） = min{ e∈E（G） f（e）｜f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st （G）.     

The Twin Domination Numb er of Strong Pro duct of Digraphs

Let γ*(D) denote the twin domination number of digraph D and let D_1  D_2 denote the strong product of D_1 and D_2. In this paper, we obtain that the twin domination number of strong product of two directed cycles of length at least 2.Furthermore, we give a lower bound of the twin domination number of strong product of two digraphs, and prove that the twin domination number of strong product of the complete digraph and any digraph D equals the twin domination number of D.     

图的强符号圈控制数

本文研究了图的强符号圈控制数γ′_(ssc)(G).利用最大独立集最大匹配等方法,刻画了满足γ′_(ssc)(G)=|E|-2的所有连通图,给出了γ′_(ssc)(G)的一个下界,求出了两类特殊图的强符号圈控制数.     

图上Nordhaus-Gaddum型的符号全控制数的界

函数f:V(G)→{-1,1}称为图G的符号全控制函数,如果对每一个开邻域集上的点的函数值的和都大于等于1.符号全控制函数的权值是指图中所有点的函数值的求和.图的符号全控制数为图中所有符号全控制函数的最小权值.令G表示图G的补图.在该文中,我们研究符号全控制数的Nordhaus-Gaddum型不等式,给出了路与其补图的符号全控制数和的上界,以及图与其补图的符号全控制数和的下界.     

关于图的反符号圈控制数

本文研究了图的反符号圈控制的问题.利用分类和反证的方法,获得了满足反符号圈控制数为负边数加4的连通图的刻画和完全二部分图的反符号圈控制数.     

有向圈的笛卡尔积有向图的双控制数

Let γ*(D) denote the twin domination number of digraph D and let Cm Cn denote the Cartesian product of C_m and C_n, the directed cycles of length m, n ≥ 2. In this paper, we determine the exact values: γ*(C_2?C_n) = n; γ*(C_3 ?C_n) = n if n ≡ 0(mod 3),otherwise, γ*(C_3?C_n) = n + 1; γ*(C_4?C_n) = n + n/2 if n ≡ 0, 3, 5(mod 8), otherwise,γ*(C_4?C_n) = n + n/2 + 1; γ*(C_5?C_n) = 2n; γ*(C_6?C_n) = 2n if n ≡ 0(mod 3), otherwise,γ*(C_6?C_n) = 2n + 2.     

不含相邻三角形的平面图的无圈边着色

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles.The acyclic edge chromatic number of a graph G is the minimum number k such that there exists an acyclic edge coloring using k colors and is denoted by χ’ a(G).In this paper we prove that χ ’ a(G) ≤(G) + 5 for planar graphs G without adjacent triangles.     

交换环上上三角矩阵的李三导子

Let T(n,R) be the Lie algebra consisting of all n × n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T(n,R)-bimodule.In this paper,we prove that every Lie triple derivation d : T(n,R) → M is the sum of a Jordan derivation and a central Lie triple derivation.     

树的罗马控制数和控制数

A Roman dominating function on a graph G = (V, E) is a function f : V→{0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) - 2. The weight of a Roman dominating function is the value (?). The minimum weight of a Roman dominating function on a graph G, denoted byγR(G), is called the Roman dominating number of G. In this paper, we will characterize a tree T withγR(T) =γ(T) 3.     

关于给定符号圈控制数的图的刻画（英文）

设G=(V,E)是简单图.称一个函数f:E→{+1,-1}为图G的符号圈控制函数,若对G的每一导出圈C,有Σ_(e∈)E(C)f(e)≥1.G的符号圈控制数被定义为γ′_(sc)(G)=min{∑_(e∈E)f(e)|f是G的符号圈控制函数}.本文刻画了所有具有γ′_(sc)(G)=|E|-4的连通图G.     

关于分圆域及其最大实子域的类数上界

设ｍ是适合m≠2（ｍｏｄ４）的整数，(?)是ｍ次本原单位根，又设Δ_k,h_m分别是分圆域Ｋ＝Ｑ(?)的判别式和类数．本文证明了：     

Rainbow Numbers for Cycles with Pendant Edges

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f (n, H) is the maximum number of colors in an edge-coloring of K _n with no rainbow copy of H. The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of K _n with rb(n, H) number of colors contains a rainbow copy of H. Certainly rb(n, H) = f(n, H) + 1. Anti-Ramsey numbers were introduced by Erdős et al. [4] and studied in numerous papers. We show that for n ≥ k + 1, where C _k ⁺ denotes a cycle C _k with a pendant edge.     

Improved Upper Bounds for Gallai–Ramsey Numbers of Paths and Cycles

Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k‐edge coloring of contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai–Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of k.     

Homogeneous Embeddings of Cycles in Graphs

If G and H are vertex-transitive graphs, then the framing number fr(G,H) of G and H is defined as the minimum order of a graph every vertex of which belongs to an induced G and an induced H. This paper investigates fr(C _m,C _n) for m<n. We show first that fr(C _m,C _n)≥n+2 and determine when equality occurs. Thereafter we establish general lower and upper bounds which show that fr(C _m,C _n) is approximately the minimum of and n+n/m. Received: June 12, 1996 / Revised: June 2, 1997     

Rainbow Numbers for Cycles in Plane Triangulations

In the article, the existence of rainbow cycles in edge colored plane triangulations is studied. It is shown that the minimum number of colors that force the existence of a rainbow C₃ in any n‐vertex plane triangulation is equal to . For a lower bound and for an upper bound of the number is determined.     

Ramsey数R_m(3)的计算及Schur数的新上界

利用抽屉原理,给出了Ramsey数Rm(3)的一个递推公式,得到Rm(3)准确值计算的一个具体表达式,并利用Rm(3)的计算公式给出了Schur数的一个新的上界。     

几类新的上可嵌入图

讨论了几类上可嵌入的边连通简单图,得到了如下结果:若G为简单连通图,且满足以下条件1)-3)之一:1)G为1-边连通的,且不含完全图K_3,α(G)≤3,2)G为2-边连通的,且不含完全图K_3,α(G)≤5,3)G为3-边连通的,且不含完全图K_3,α(G)≤10,则G是上可嵌入的,且在上述相应条件下,独立数上界都分别是最好的.     

The Ramsey Numbers of Large cycles Versus Odd Wheels

For given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(C_n, W_m) = 3n − 2 for odd m ≥ 5 and . Surahmat, Ioan Tomescu: Part of the work was done while the first and the last authors were visiting the School of Mathematical Sciences, Government College University, Lahore, Pakistan. Surahmat: Research partially support under TWAS, Trieste, Italy, RGA No: 06-018 RG/MATHS/AS–UNESCO FR: 3240144875.     

图的下测地数和上测地数

对于图G(或有向图D)内的任意两点u和v,u—v测地线是指在u和v之间(或从u到v)的最短路．I(u,v)表示位于u—v测地线上所有点的集合,对于S(?)V(G)(或V(D)),I(S)表示所有I(u,v)的并,这里u,v∈S．G(或D)的测地数g(G)(或g(D))是使I(S)=V(G)(或I(S)=V(D))的点集S的最小基数．G的下测地数g~-(G)=min{g(D)：D是G的定向图},G的上测地数g~ (G)=max{g(D)：D是G的定向图}．对于u∈V(G)和v∈V(H),G_u H_v表示在u和v之间加一条边所得的图．本文主要研究图G_u H_v的测地数和上(下)测地数．     

Triple Crossing Numbers of Graphs

We introduce the triple crossing number,a variation of the crossing number,of a graph,which is the minimal number of crossing points in all drawings of the graph with only triple crossings.It is defined to be zero for planar graphs,and to be infinite for non-planar graphs which do not admit a drawing with only triple crossings.In this paper,we determine the triple crossing numbers for all complete multipartite graphs which include all complete graphs.