自然数列对虾树及其串联树的强优美性

具有完美匹配M的n阶树T是强优美的,如果对任意的uv∈M,存在树T的一个优美标f,使得f(u)+f(v)=n-1.讨论了自然数列对虾树及其串联树的强优美标号.     

探索金鱼图的奇优美性

给出了奇优美图和二分奇优美图的概念,并定义了金鱼图,证明了在鱼头为不同图形的情况下,金鱼图仍然是奇优美的,且是二分奇优美的.还证明了:对一个奇优美图H和一棵二分奇优美树T,用一条边连接T的一个顶点和H的标号基点u_0后所得到的金鱼图仍是奇优美图.     

奇优美图的构造

图G的标号是指G的顶点集到一个整数集的映射f,且对e=uv∈E(G)由f(u)和f(v)诱导出边e的标号f(uv).本文给出了二分奇优美图的概念,证明了一个图是二分奇优美的当且仅当它是二分优美的,并给出了一些构造奇优美图的方法.     

(n,m)-p-蜘蛛树的强优美性

给出了优美树、强优美树、边对称树以及对偶标号的概念,定义了一类蜘蛛树.证明了此类蜘蛛树是强优美树,蜘蛛树的强优美标号是对偶标号,并证明了蜘蛛树的边对称树仍然是强优美树.     

(2m+1,1)-p-树的二分强优美性和二分强奇优美性

给出了二分优美树和强优美树、强奇优美树、边对称树以及对偶标号的概念,定义了一类(2m+1,1)-p-树.并证明了(2m+1,1)-p-树是二分强优美树和二分强奇优美树,并验证了(2m+1,1)-p-树的优美标号是对偶标号.最后证明了(2m+1,1)-p-树的边对称树仍然是二分驺优美树,并将上面的结论推广到一般情形.     

关于P_(r,(2s-1))的奇优美标号

对于简单图G=〈V,E〉,如果存在一个映射f:V(G)→{0,1,2,…,2 |E|-1}满足1)对任意的u,v∈V,若u≠v,则(u)≠f(v);2)max{f(v)|v∈V}=2|E|-1;3)对任意的e_1,e_2∈E,若e_1≠e_2,则g(e_1)≠g(e_2),此处g(e)=|f(u)+f(v)|,e=uv;4){g(e)|e∈E}={1,3,5,…,2|E|-1},则称G是奇优美图,f称为G的奇优美标号.Gnanajoethi提出了一个猜想:每棵树都是奇优美的.证明了图P_(r,(2s-1)是奇优美图.     

一类优美树

§1. Preliminary Lemma Using A to denote the cardinal number of a finite set A, we do not explain anymore what are similar to A in this paper. For a given tree T(V,E), if there is a labeling f of its vertices, which satisfiesf［V(T)］={f(u)u∈V(T)}={0,1,2,…,E(T)},lf［E(T)］={lf(uv)=f(u)-f(v), uv∈E(T)}={1,2,…,E(T)},then f is called a graceful labeling of T, and T is called a graceful tree. lf denotes the labeling of edges that is derived from f. In the 1960s, RingedKotzing and A. R…     

几类有趣图的奇优美性和奇强协调性

对于简单图G=〈V,E〉,如果存在一个映射f:V(G)→{0,1,2,…,2|E|-1}满足:1)对任意的u,v∈V,若u≠v,则f(u)≠f(v);2)max{f(v)|v∈V}=2|E|-1;3)对任意的e_1,e_2∈E,若e_1≠e_2,则g(e_1)≠g(e_2),此处g(e)=|f(u)+f(v)|,e=uv;4)|g(e)|e∈E}={1,3,5,…,2|E|-1},则称G为奇优美图,f称为G的奇优美标号.设G=〈V,E〉是一个无向简单图.如果存在一个映射f:V(G)→{0,1,2,…,2|E|-1},满足:1)f是单射;2)■uv∈E(G),令f(uv)=f(u)+f(v),有{f(uv)|uv∈E(G)}={1,3,5,…,2|E|-1},则称G是奇强协调图,f称为G的.奇强协调标号或奇强协调值.给出了链图、升降梯等几类有趣图的奇优美标号和奇强协调标号.     

优美图的一些性质

对于一个(p,q)-图G,如果存在一个单射.f:V(G)→{0,1,…,q},使得边标号集合{f(uv)| uv∈E(G)}={1,2,…,q},其中边标号为f(uv)=|f(u)-f(v)|,那么称G是优美图,并称.f是G的一个优美标号.通过研究若干优美图,得出一些优美图的性质.     

伪完全二分图PK_(n,n)奇优美标号的计算机实现

给出了伪完全二分图PK_(n,n)的定义及性质,提出了该类图的奇优美标号算法,证明了算法的正确性及时间复杂度,从而证明了伪完全二分图的奇优美性.并给出了伪完全二分图PK_(n,n),当n=3,4,5的一种标号方法.     

具有匹配对网络模型的优美性

在当今网络研究中,人们需要将某些特殊的图分解为指定的结构.优美图可以被运用到图分解中.得到一些构造优美图的可算法化的方法,并构造较为复杂的优美图.     

Almost all trees are almost graceful

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled by using the numbers {1,2,…,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each γ>0 and for all n>n₀(γ). Suppose that (i) the maximum degree of T is bounded by ), and (ii) the vertex labels are chosen from the set {1,2,…,?(1+γ)n?}. Then there is an injective labeling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability.     

Decompositions of graphs via generalized graceful labelings

In [Pasotti, A., On d-graceful labelings, to appear on Ars Combin] a d-divisible α-labeling is defined as a generalization of the classical one of Rosa (see [Rosa, A., On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967), 349–355]) and, similarly to the classical case, it is proved that there exists a link between d-divisible α-labelings of a graph Γ and cyclic Γ-decompositions. In [Benini, A., and Pasotti, A., Decompositions of complete multipartite graphs via generalized graceful labelings, submitted. (arXiv:1210.4370)] we have dealt with the existence of d-divisible α-labelings of caterpillars and certain classes of cycles and hairy cycles and the resulting possible decompositions.     

关于P_n~3的优美性(英文)

设G(V,E)是一个简单图,对自然数k,当V(G~k)=V(G,E(G~k)=E(G)∪{uv|d(u,v)=k},则称图G~k为k-次方图,本文证明了图P_n~3的优美性。     

关于笛卡尔乘积图的优美性

研究了笛卡尔乘积图Pm×Pn×P1的优美标号算法,并且给出了他们都是优美图的证明,同时推广了笛卡尔乘积图Pm×Pn是优美图的结论.     

一类偶阶图的边优美性

本文讨论了一类偶阶图的边优美性。同时得到了完全图是边优美图的充要条件。     

关于有向图n·_3是优美图的一个猜想的证明

在文 [1 ]中提出猜想 :当 n≡ 0 (mod2 )时 ,n· C 3是优美图 .本文证明了这个猜想 .