整数距离图G(Dm,k,2)的点线性荫度

整数距离图G(D)以全体整数作为顶点集,顶点u,v相邻当且仅当|u-v|∈D,其中D是一个正整数集．本文讨论整数距离图的点线性荫度,记为vla(G(D))．对于m≥5k,设D_(m,k,2)={1,2,…,m}/{k,2k),得到vla(G(D_(m,1,2)))=■并决定出了G(D_(m,2,2))在某些特殊的仇值上点线性荫度的确切值以及当k≥3时G(D_(m,k,2))的点线性荫度的上、下界．     

退化图中的子图计数问题研究

令G表示n个顶点的图,如果G的每个子图中都包含一个度至多为k的顶点,则称G为k-退化图.令N(G,F)表示G中F子图的个数.主要研究了k-退化图中完全子图和完全二部子图的计数问题,给出了计数的上界以及相应的极图.首先,证明了Ν(G,K_t)≤(n-k)(k t-1)+(k t).其次,如果s,t≥1,n≥k+1且s+t≤k,我们证明了Ν(G,K_s,t)≤{(k s)(n-s s)-1/2(k s)(k-s s),t=s,(k s)(n-s t)+(k t)(n-t s)-(k t)(k-t s),t≠s.此外,还研究了在最大匹配和最小点覆盖为给定值的情况下,图G中的最大边数.记v(G),K(G)分别为图G的最大匹配数和最小点覆盖.证明了当v(G)≤k,K(G)=k+r且n≥2k+2r²+r+1时,有e(G)≤(k+r+1 2)+(k-r)(n-k-r-1).     

关于Erds-Jacobson-Lehel问题的门槛(英文)

设σ(k ,n)表示最小的正整数m ,使得对于每个n项正可图序列 ,当其项和至少为m时 ,有一个实现含k+ 1个顶点的团作为其子图 .Erd s等人猜想 :σ(k ,n) =(k - 1 ) ( 2n-k)+ 2 .Li等人证明了这个猜想对于k≥ 5,n≥ k2 + 3是对的 ,并且提出如下问题 :确定最小的整数N(k) ,使得这个猜想对于n≥N(k)成立 .他们同时指出 :当k≥ 5时 ,5k- 12 ≤N(k)≤ k2 + 3.Mubayi猜想 :当k≥ 5时 ,N(k) =5k - 12 .在本文中 ,我们证明了N( 8) =2 0 ,即Mubayi猜想对于k =8是成立的     

平面图点荫度的一个局部条件

图G的点荫度va(G)是指V(G)的最小划分数,使得每个点划分集的导出子图是一个森林.众所周知,平面图G的点荫度至多为3.2008年,Raspaud和王维凡证明了:若G是不含κ-圈(κ∈{3,4,5,6})的平面图,则va(G)≤2.本文推广了上述结果,得到了点荫度至多为2的一个局部条件,即若平面图G的每一个点都不同时与3-圈、4-圈、5-圈和6-圈关联,则va(G)≤2.     

完全多部图K_(1*r,3*(k-2))的m数

如果对一个图G的每个顶点v,任给一个k-列表L(v),使得G要么没有正常列表染色,要么至少有两种正常列表染色,则称图G具有M(k)性质.定义图G的m数为使得图G具有M(k)性质的最小整数k,记为m(G).已有研究表明,当k=3,4时,图K_(1*r,3*(k-2))具有M(k)性质,且当r≥2时,m(K_(1*r,3*(k-2)))=k.本文将上述结论推广到每一个k,证明了对任意r∈N~+,k≥3,图K_(1*r,3*(k-2))具有M(k)性质,且当k≥4,r≥(k-2)时,m(K_(1*r,3*(k-2)))=k.此外,得到图K_(1,3,3,3)的m数为4,该图是图K_(1*r,3*(k-2))中r=1,k=5时的特殊情况,同时也是现有研究中尚未解决的一个问题.     

正则多部竞赛图的竞争指数

设D是一个有向图,若存在无向图G满足:(1)G的顶点集与D的顶点集相同;(2)任取D中的两个顶点x,y,其在G中相邻当且仅当存在D中顶点z,使得D中包含一条从x到z的长为m的有向途径和一条从y到z的长为m的有向途径,则称G为D的m步竞争图,记为G=C~m(D).2004年,Cho和Kim首次提出竞争指数的概念.若对于某个正整数r和所有非负整数i,存在最小正整数q,使C~(q+i)(D)=C~(q+i+r)(D),则称整数q为D的竞争指数,记为cindex(D).2008年,Kim给出了竞赛图的竞争指数的上界.2009年,Akelbek和Kirkland给出了本原有向图的竞争指数.文中研究并计算了正则多部竞赛图的竞争指数.     

随机图的广义3-连通度

图的广义连通度的概念是由Chartrand等人引入的.令S表示图G的一个非空顶点集,κ(S)表示图G中连结S的内部不交树的最大数目.那么,对任意一个满足2≤r≤n的整数r,定义G的广义r-连通度为所有κ(S)中的最小值,其中S取遍G的顶点集合的r-元子集.显然,κ_2(G)=κ(G),即为图G的顶点连通度.所以广义连通度是经典连通度的一个自然推广.本文研究了随机图的广义3-连通度,证明了对任一给定的整数k,k≥1,p=(log n+(k+1)log long n-log lon logn)/n是关于性质κ_3(G(n,p))≥k的紧阈值函数.我们得到的结果可以看作是Bollobas和Thomason给出的关于经典连通度结果的推广.     

强定向图的k-强距离

对强连通有向图D的一个非空顶点子集S,D中包含S的具有最少弧数的强连通有向子图称为S的Steiner子图,S的强Steiner距离d(S)等于S的Steiner子图的弧数. 如果|S|=k, 那么d(S)称为S的k-强距离. 对整数k≥2和强有向图D的顶点v,v的k-强离心率sek(v)为D中所有包含v的k个顶点的子集的k-强距离的最大值. D中顶点的最小k-强离心率称为D的k-强半径,记为sradk(D),最大k-强离心率称为D的k-强直径,记为sdiamk(D). 本文证明了,对于满足k+1≤r,d≤n的任意整数r,d,存在顶点数为n的强竞赛图T′和T″,使得sradk(T′)=r和sdiamk(T″)=d;进而给出了强定向图的k-强直径的一个上界.     

偶数阶W(4,n)的κ-边优美的图标号

设k是一个非负整数,G是一个p点q边图.如果将G的边用k,k+1,k+2,…,k+q-1进行标号,而顶点标号模p运算后各不相同,那么称图G是后一边优美的.记EGI(G)是所有满足G是k-边优美的k的集合,称EGI(G)是G的边优美指标集.主要是研究n为偶数时W(4,n)的边优美指标集.     

五类特殊图的三种距离矩阵的特征多项式

设G是一个具有顶点集V(G)={v_1,v_2,…,u_n}的n阶简单图.设d_(i,j)=d(v_i,v_j)表示图G中任意两个顶点v_i与v_j的距离.矩阵D(G)=[d_(i,j)]_(n×n)定义为图G的距离矩阵.定义Tr(v)=∑_(ueV(G))d(u,u)为图G中顶点u的点传递度.Diag(Tr)表示以G中顶点的点传递度为主对角线上元素的对角矩阵.则矩阵D~L(G)=Diag(Tr)一D(G)和D~Q(G)=Diag(Tr)+D(G)分别定义为图G的距离拉普拉斯矩阵和距离无符号拉普拉斯矩阵.分别得到五类特殊图的距离,距离拉普拉斯,距离无符号拉普拉斯的特征多项式的一般表达式.     

A lower bound for the distance k-domination number of trees

A subset D of vertices of a graph G = (V, E) is a distance k-dominating set for G if the distance between every vertex of V ? D and D is at most k. The minimum size of a distance k-dominating set of G is called the distance k-domination number γ_k(G) of G. In this paper we prove that (2k + 1)γ_k(T) ≥ ¦V¦ + 2k ? kn₁ for each tree T = (V, E) with n₁ leafs, and we characterize the class of trees that satisfy the equality (2k + 1)γ_k(T) = ¦V¦ + 2k ? kn₁. Our results generalize those of Lemanska [4] for k = 1 and of Cyman, Lemanska and Raczek [1] for k = 2.     

A bound on the <Emphasis Type="Italic">k</Emphasis>-domination number of a graph

Let G be a graph with vertex set V(G), and let k ⩾ 1 be an integer. A subset D ⊆ V(G) is called a k-dominating set if every vertex υ ∈ V(G)-D has at least k neighbors in D. The k-domination number γ _k (G) of G is the minimum cardinality of a k-dominating set in G. If G is a graph with minimum degree δ(G) ⩾ k + 1, then we prove that

完全图的Cantesian积的L(2,1)-圆标定

For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L（j, k）-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L（j, k）-labelings of G. An m-（j, k）-circular labeling of a graph G is a function f ： V（G） →{0, 1, 2,..., m - 1} such that ｜f（u） - f（v）｜m ≥ j if u and v are adjacent; and ｜f（u） - f（v）｜m 〉 k ifu and v are at distance two, where ｜x｜m = min{｜xl｜, m-｜x｜}. The minimum integer m such that there exists an m-（j, k）-circular labeling of G is called the σj,k-number of G and is denoted by σj,k（G）. This paper determines the σ2,1-number of the Cartesian product of any three complete graphs.     

Almost-regular factorization of graphs

For integers a and b, 0 ? a ? a ? b, an [a, b]-graph G satisties a ? deg(x, G) ? b for every vertex x of G, and an [a, b]-factor is a spanning subgraph F such that a ? deg(x, F) ? b for every vertex x of F. An [a, b]-factor is almost-regular if b = a + 1. A graph is [a, b]-factorable if its edges can be decomposed into [a, b]-factors. When both K and t are positive integers and s is a nonnegative integer, we prove that every [(12K + 2)t + 2ks, (12k + 4)t + 2ks]-graph is [2k,2k + 1]-factorable. As its corollary, we prove that every [r.r + 1]-graph with r ? 12k² + 2k is [2k + 1]-factorable, which is a partial extension of the two results, one by Thomassen and the other by Era.     

The vertex linear arboricity of distance graphs

A distance graph is a graph G(R,D) with the set of all points of the real line as vertex set and two vertices u,v∈R are adjacent if and only if |u-v|∈D where the distance set D is a subset of the positive real numbers. Here, the vertex linear arboricity of G(R,D) is determined when D is an interval between 1 and δ. In particular, the vertex linear arboricity of integer distance graphs G(D) is discussed, too.     

GENERALISED MAXIMAL INDEPENDENCE AND CLIQUE NUMBERS OF GRAPHS

Abstract

A set B of vertices of a graph G = (V,E) is a k-maximal independent set (kMIS) if B is independent but for all ?-subsets X of B, where ? ? k—1, and all (? + 1)-subsets Y of V—B, the set (B—X) u Y is dependent. A set S of vertices of C is a k-maximal clique (kMc) of G iff S is a kMIS of [Gbar]. Let β_k, (G) (w_k(G) respectively) denote the smallest cardinality of a kMIS (kMC) of G—obviously β_k(G) = w_k([Gbar]). For the sequence m₁ ? m₂ ?…? m_n = r of positive integers, necessary and sufficient conditions are found for a graph G to exist such that w_k(G) = m_k for k = 1,2,…,n and w(G) = r (equivalently, β_k(G) = m_k for k = 1,2,…,n and β(G) = r). Define s_k(?,m) to be the largest integer such that for every graph G with at most s_k(?,m) vertices, β_k(G) ? ? or w_k(G) ? m. Exact values for s_k(?,m) if k ≥ 2 and upper and lower bounds for s₁(?,m) are de termined.     

Bounds on a generalized domination parameter

Abstract

If n is an integer, n ≥ 2 and u and v are vertices of a graph G, then u and v are said to be K_n-adjacent vertices of G if there is a subgraph of G, isomorphic to K_n , containing u and v. For n ≥ 2, a K_n- dominating set of G is a set D of vertices such that every vertex of G belongs to D or is K_n-adjacent to a vertex of D. The K_n-domination number γK_n (G) of G is the minimum cardinality among the K_n-dominating sets of vertices of G. It is shown that, for n ε {3,4}, if G is a graph of order p with no K_n-isolated vertex, then γK_n (G) ≤ p/n. We establish that this is a best possible upper bound. It is shown that the result is not true for n ≥ 5.     

既不含4-圈又不含6-圈的平面图的非正常染色

设d₁, d₂,..., d_k是k个非负整数. 若图G=(V,E)的顶点集V能被剖分成k个子集V₁, V₂,...,V_k, 使得对任意的i=1, 2,..., k, V_i的点导出子图G[V_i] 的最大度至多为d_i, 则称图G是(d₁, d₂,...,d_k)-可染的. 本文证明既不含4-圈又不含6-圈的平面图是(3, 0, 0)-和(1, 1, 0)-可染的.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.