$P_m\times K_n$的邻点可区别全色数

设 $G$ 是简单图. 设$f$是一个从$V(G)\cup E(G)$ 到$\{1, 2,\cdots, k\}$的映射. 对每个$v\in V(G)$, 令 $C_f (v)=\{f(v)\}\cup \{f(vw)|w\in V(G), vw\in E(G)\}$. 如果 $f$是$k$-正常全染色, 且对任意$u, v\in V(G), uv\in E(G)$, 有$C_f(u)\ne C_f(v)$, 那么称 $f$ 为图$G$的邻点可区别全染色(简称为$k$-AVDTC).数 $\chi_{at}(G)=\min\{k|G$ 有$k$-AVDTC\}称为图$G$的邻点可区别全色数.本文给出路$P_m$和完全图$K_n$ 的Cartesion积的邻点可区别全色数.     

三类有向三元系之间关系的一种猜想

一个Mendelsohn (Directed, 或Hybrid)三元系 MTS$(v, \lambda)$~(DTS$(v, \lambda)$,或HTS$(v,\lambda))$, 是由$v$元集$X$ 上的一些循环（可迁，或循环和可迁）三元组(简称区组)构成的集合${\cal B}$, 使得$X$上每个由不同元素组成的有序对都恰在 ${\cal B}$的$\lambda$个区组中出现.本文主要讨论了这三类有向三元系之间的一种关联关系，给出猜想：任意MTS$(v,\lambda)$的区组关联图$G(\     

不含某些子图的k连通图中的k可收缩边

最近Ando等证明了在一个$k$($k\geq 5$ 是一个整数) 连通图 $G$ 中,如果 $\delta(G)\geq k+1$, 并且 $G$ 中既不含 $K^{-}_{5}$,也不含 $5K_{1}+P_{3}$, 则$G$ 中含有一条 $k$ 可收缩边.对此进行了推广,证明了在一个$k$连通图$G$中,如果 $\delta(G)\geq k+1$,并且 $G$ 中既不含$K_{2}+(\lfloor\frac{k-1}{2}\rfloor K_{1}\cup P_{3})$,也不含 $tK_{1}+P_{3}$ ($k,t$都是整数,且$t\geq 3$),则当 $k\geq 4t-7$ 时, $G$ 中含有一条 $k$ 可收缩边.     

图加一条边后的带宽和

令$BS(G,f)=\sum\limits_{uv\in E(G)}|f(u)-f(v)|$, 其中$f$为$V(G)\rightarrow\{1,2,\cdots,|V(G)|\}$的双射, 并称$BS(G)=\min\limits_{f}BS(G,f)$为图$G$的带宽和. 讨论顶点数为$n$的简单图$G$加上一条边$e\in\overline{E(G)}$后, 带宽和$BS(G+e)$与$BS(G)$的关系, 得其关系式$BS(G)+1\leq BS(G+e)\leq BS(G)+n-1$. 并证明此不等式中等号可取到, 即存在图$G_{1}$和$G_{2}$使得$BS(G_{1}+e)=BS(G_{1})+1$, $BS(G_{2}+e)=BS(G_{2})+n-1$.     

偶图$K_{n,n+7}-A(|A|\leq3)$的圈长分布唯一性

阶为$n$的图$G$的圈长分布是序列($c_1,c_2,\ldots,c_n$), 其中$c_i$是图$G$中长为$i$的圈数.本文得到如下结果: 设$A\subseteq E(K_{n,n+7})$,在以下情况, 图 $G$ 由其圈长分布唯一确定.(1) $G=K_{n,n+7}$(n\geq10)$;(2) $G=K_{n,n+7}-A$ $(|A|=1,n\geq12)$;(3)$G=K_{n,n+7}-A$(|A|=2,n\geq14)$;(4)$G=K_{n,n+7}-A$ $(|A|=3     

一个关于平面图的K-(2,1)-全可选性的结果

设图$G$的一个列表分配为映射$L: V(G)\bigcup E(G)\rightarrow2^{N}$. 如果存在函数$c$使得对任意$x\in V(G)\cup E(G)$有$c(x)\in L(x)$满足当$uv\in E(G)$时, $|c(u)-c(v)|\geq1$, 当边$e_{1}$和$e_{2}$相邻时, $|c(e_{1})-c(e_{2})|\geq1$, 当点$v$和边$e$相关联时, $|c(v)-c(e)|\geq 2$, 则称图$G$为$L$-$(p,1)$-全可标号的. 如果对于任意一个满足$|L(x)|=k,x\in V(G)\cup E(G)$的列表分配$L$来说, $G$都是$L$-$(2,1)$-全可标号的, 则称$G$是 $k$-(2,1)-全可选的. 我们称使得$G$为$k$-$(2,1)$-全可选的最小的$k$为$G$的$(2,1)$-全选择数, 记作$C_{2,1}^{T}(G)$. 本文, 我们证明了若$G$是一个$\Delta(G)\geq 11$的平面图, 则$C_{2,1}^{T}(G)\leq\Delta+4$.     

增益图与其底图的正惯性指数之间的关系

设$\mathbb{T}$是模为1的复数乘法子群.图$G=(V,E)$,这里$V,E$分别表示图的点和边.增益图是将底图中的每条边赋于$\mathbb{T}$中的某个数值$\varphi(v_iv_j)$,且满足$\varphi(v_iv_j) =\overline{\varphi(v_jv_i)}$.将赋值以后的增益图表示为$(G,\varphi)$.设$i_+(G,\varphi)$和$i_+(G)$分别表示增益图与底图的正惯性指数,本文证明了如下结论: $$ - c( G ) \le {i_ + } ( {G,\varphi } ) - {i_ + }( G ) \le c( G ), $$ 这里$c(G)$表示圈空间维数,并且刻画了等号成立时候的所有极图.     

图$P_m\times P_n, P_m\times C_n$ 和C_m\times C_n$}的邻点可区别全色数的注记

设 $G$ 是一个简单图. 设$f$是从$V(G) \cup E(G)$到 $\{1, 2,\ldots, k\}$的一个映射.对任意的 $v\in V(G)$, 设$C_f(v)=\{f(v)\}\cup \{f (vw)|w\in V(G),vw\in E(G)\}$ . 如果 $f$ 是一个 $k$-正常全染色, 且对 $u, v\in V(G),uv\in E(G)$, 有 $C_f(u)\neq C_f(v)$, 那么称 $f$ 为$k$-邻点可区别全染色 (简记为$k$-$AVDTC$). 设     

半无爪可迹图的度和条件

可迹图即为一个含有Hamilton路的图.令$N[v]=N(v)\cup\{v\}$, $J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}$.若图中任意距离为2的两点$u,v$满足$J(u,v)\neq \emptyset$,则称该图为半无爪图.令$\sigma_{k}(G)=\min\{\sum_{v\in S}d(v):S$为$G$中含有$k$个点的独立集\},其中$d(v)$表示图$G$中顶点$v$的度.本论文证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{3}(G)\geq {n-2}$,则图$G$为可迹图; 文中给出一个图例,说明上述结果中的界是下确界; 此外,我们证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{2}(G)\geq \frac{2({n-2})}{3}$,则该图为可迹图.     

有限群的$\Phi$-$\tau$-可补子群

假设$\tau$是一个子群算子, $H$是有限群$G$的一个$p$-子群. 令 $\bar{G}=G/H_{G}$且$\bar{H}=H/H_{G}$, 如果$\bar{G}$有一个次正规子群$\bar{T}$ 和一个包含于$\bar{H}$ 的$\tau$-子群$\bar{S}$满足$\bar{G}=\bar{H}\bar{T}$且$\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})$, 就称$H$是$G$的一个$\Phi$-$\tau$- 可补子群. 文章通过讨论群$G$的准素数子群的$\Phi$-$\tau$-可补性给出了超循环嵌入和$p$-幂零性的一些新的特征.     

2类6点7边图的$\lambda$-填充与$\lambda$-覆盖

A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi, λ）-OPD （OCD） for v ≡ 2, 3, 4, 5, 6 （mod 7）, λ ≥ 1, i = 1, 2.     

$K_v$的最大$K_2K_3$填充设计

Abstract Let Kv be the complete graph on v vertices, and G a finite simple undirected graph without isolated vertices. A G-packing of Kv, denoted by （v, G, 1）-packing, is a pair （X,A） where X is the vertex set of K＋ and ＋4 is a family of edge-disjoint subgraphs isomorphic to G in Kv. In this paper, the maximum number of subgraphs in a （v, G, 1）-packing is determined when G is K2 x K3, the Cartesian product of K2 and K3, leaving two orders undetermined. This design originated from the use of DNA library screening.     

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.     

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.     

$k$-广义拟树的第一个leap Zagreb 指标

图$G$的第一个leap Zagreb指标定义如下: $LM_1(G)=\sum_(v\in v(G)}d_2(v/G)^2$, 其中$d_2(v/G)$是离点$v$的距离为2的顶点. 令$\mathcal{QT}^{(k)}(n)$是有$n$个顶点的$k$-广义拟树的集合.若$G\in \mathcal{QT}^{(k)}(n)$, 本文给出了图$G$的第一个leap Zagreb指标的范围.     

$\alpha$-可分解4-圈系统

An m-cycle system of order v and index λ, denoted by m-CS（v,λ）, is a collection of cycles of length m whose edges partition the edges of λKv. An m-CS（v,λ） is α-resolvable if its cycles can be partitioned into classes such that each point of the design occurs in precisely α cycles in each class. The necessary conditions for the existence of such a design are m｜λv（v-1）/2,2｜λ（v -1）,m｜αv,α｜λ（v-1）/2. It is shown in this paper that these conditions are also sufficient when m = 4.     

关于完全二部图$K_{4, n}$和$K_{n, n}$的点可区别 $IE$-全染色的一些注记

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

树的导出匹配覆盖

The induced matching cover number of a graph G without isolated vertices,denoted by imc(G),is the minimum integer k such that G has k induced matchings M1,M2,…,Mk such that,M1∪M2 ∪…∪Mk covers V(G).This paper shows if G is a nontrivial tree,then imc(G) ∈ {△*0(G),△*0(G) + 1,△*0(G)+2},where △*0(G) = max{d0(u) + d0(v) :u,v ∈ V(G),uv ∈ E(G)}.     

以PSL(2,q)作为本原自同构群的旗传递对称$(v,k,\lambda)$设计的分类

设$D$是一个非平凡的对称$(v,k,\lambda)$设计, $G$是$D$的一个自同构群.本文证明了如果$G$以二维典型群PSL$(2,q)$作为基柱且在$D$上的作用是旗传递和点本原的,那么设计$D$的参数只能为$(7, 3, 1)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 3)$或$(15, 8, 4)$.