图的好符号星控制数

引入了图的好符号星控制的概念,求出了欧拉图、完全二部图、完全图和轮图的好符号星控制数,并改进了图的符号星控制数的两个上界.     

图的三种符号星控制数

引入了图的符号星k限定控制的概念,从而求出了星图和轮图的符号星k控制数.还刻画了满足γ′_(ss)(G)=1/2(2r+s)的图,基中γ′_(ss)(G)表示图G的符号星控制数.最后对图的符号星部分控制的已有结果作了改进.     

图的符号星k控制数

引入了图的符号星k控制的概念.设G=（V,E）是一个图,一个函数f：E→{-1,＋1},如果∑e∈E[v]f（e）≥1对于至少k个顶点v∈V（G）成立,则称f为图G的一个符号星k控制函数,其中E（v）表示G中与v点相关联的边集.图G的符号星k控制数定义为γkss（G）=min{∑e∈Ef（e）｜f为图G的符号星k控制函数}.在本文中,我们主要给出了一般图的符号星k控制数的若干下界,推广了关于符号星控制的一个结果,并确定路和圈的符号星k控制数.     

图的符号星部分控制数

引入了图的符号星部分控制的概念.设G=(V,E)是一个简单连通图, M是V的一个子集.一个函数f:E→{-1,1}若满足∑e∈E(v)f(e)≥1对M中的每个顶点v都成立,则称f是图G的一个符号星部分控制函数,其中E(v)表示G中与v点相关连的边集.图G的符号星部分控制数定义为γM(85)(G)=min{∑e∈Ef(e)|f是G的符号星部分控制函数}.在本文中我们主要给出了一般图的符号星部分控制数的上界和下界,并确定了路、圈和完全图的符号星部分控制数的精确值.作为我们引入的这一新概念的一个应用,求出了完全图的符号星k控制数.     

关于图的符号星控制数

设G=(V,E)是一个图,u∈V,则E(u)表示u点所关联的边集.一个函数f:E→{-1,1}如果满足■f(e)≥1对任意v∈V成立,则称f为图G的一个符号星控制函数,图G的符号星控制数定义为γ'_(ss)(G)=min{■f(e):f为图G的一个符号星控制函数}.给出了几类特殊图的符号星控制数,主要包含完全图,正则偶图和完全二部图.     

特殊图类的符号控制数

图G的符号控制数γS(G)有着许多重要的应用背景.已知它的计算是NP-完全问题,因而确定其上下界有重要意义.本文研究了1)一般图G的符号控制数,给出了一个新的下界;2)确定了Cn图的符号控制数的精确值.     

几类图的符号控制

设G=(V,E)是一个图,一个函数f:V→{-1,+1}如果满足Σv∈N[υ]f(ν)≥1对于每个点u∈V成立,则称f为图G的一个符号控制函数,图G的符号控制数γs(G)定义为γs(G)=min{Σv∈vf(v)|f为图G的符号控制函数},类似地,可定义图G的上符号控制数Γs(G).研究了几类特殊图的符号控制问题,获得了完全l等部图和乘积图P_3×P_n的符号控制数,并确定了P_2×P_n和P_3×P_n的上符号控制数.     

关于图的符号控制数的下界

图的符号控制数的研究有许多应用背景，但图的符号控制数的计算是NP完全问题，因而确定其上下界有重大意义。本文在[5]的基础上，引进了新参数δ^*(G)，全面改进了[5]所给出的符号控制数的下界，并给出了一些可达下界的图。     

关于图的强符号全控制数

图的强符号全控制数有着许多重要的应用背景,因而确定其下界有重要的意义.本文提出了图的强符号全控制数的概念,在构造适当点集的基础上对其进行了研究,给出了：（1）一般图的强符号全控制数的5个独立可达的下界及达到其界值的图;（2）确定了圈、轮图、完全图、完全二部图的强符号全控制数的值.     

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.     

Graceful signed graphs: II. The case of signed cycles with connected negative sections

In our earlier paper [9], generalizing the well known notion of graceful graphs, a (p, m, n)-signed graph S of order p, with m positive edges and n negative edges, is called graceful if there exists an injective function f that assigns to its p vertices integers 0, 1,...,q = m + n such that when to each edge uv of S one assigns the absolute difference |f(u)-f(v)| the set of integers received by the positive edges of S is {1,2,...,m} and the set of integers received by the negative edges of S is {1,2,...,n}. Considering the conjecture therein that all signed cycles Z_k, of admissible length k 3 and signed structures, are graceful, we establish in this paper its truth for all possible signed cycles of lengths 0, 2 or 3 (mod 4) in which the set of negative edges forms a connected subsigraph.     

Shorter signed circuit covers of graphs

A signed circuit is a minimal signed graph (with respect to inclusion) that admits a nowhere-zero flow. We show that each flow-admissible signed graph on edges can be covered by signed circuits of total length at most , improving a recent result of Cheng et al. To obtain this improvement, we prove several results on signed circuit covers of trees of Eulerian graphs, which are connected signed graphs such that removing all bridges results in a collection of Eulerian graphs.     

关于图符号的边控制 (英)

设γ's(G)和γ'ι(G)分别表示图G的符号边和局部符号边控制数,本文主要证明了:对任何n阶图G(n≥4),均有γ's(G)≤[11/6n-1]和γ'ι(G)≤2n-4成立,并提出了若干问题和猜想．     

Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.     

A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V(G)and an edge set E(G). A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1}.The weight of f is W(f)=∑_(x∈V)(G)∪E(G))f(X).For an element x∈V(G)∪E(G),we define f[x]=∑_(y∈NT[x])f(y).A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1} such that f[x]≥1 for all x∈V(G)∪E(G).The total signed domination numberγ_s~*(G)of G is the minimum weight of a total signed domination function on G. In this paper,we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values ofγ_s~*(G)when G is C_n and P_n.     

Lower bounds on signed edge total domination numbers in graphs

The open neighborhood N _G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values , taken over all signed edge total dominating function f of G, is called the signed edge total domination number of G and is denoted by γ _st ′(G). Obviously, γ _st ′(G) is defined only for graphs G which have no connected components isomorphic to K ₂. In this paper we present some lower bounds for γ _st ′(G). In particular, we prove that γ _st ′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γ _st ′(T). Research supported by a Faculty Research Grant, University of West Georgia.