直径为4的奇优美树

Odd graceful trees whose diameters are four

对于简单图G=<V,E>,如果存在一个映射f:V→{0,1,2,…,2E|-1}满足:对任意的u,v∈V,若u≠v,则f(u)≠f(v);max{f(v)|v∈V}=2|E|-1;对任意的e1,e2∈E,若e1≠e2,则g(e1)≠g(e2),此处g(e)=|f(u)-f(v)|,e=uv;{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。提出一个猜想:每棵树都是奇优美的,文章证明了直径为4的树都是奇优美的。

Let G=〈V,E〉 be a simple graph. If there exists a mapping f： V→{0,1,2,…,2E｜-1}which satisfies： arbitaryu,v∈V, if u≠v,then f（u）≠f（v） { max{f（v）｜v∈V} =2 ｜E｜-1; arbitarye1 ,e2 ∈E,if e1≠e2,theng（e1）≠g（e2） ,hereg（e）=｜f（u）-f（v）｜,e=uv; {g（e） ｜e≠E}={1,3,5, …,2 ｜E｜-1},then G is called an odd graceful graph, and f is called odd graceful labeling of G. Mr. Gnanajoethi had one idea that every tree is odd graceful. In this paper, it is proved that all trees whose diameter are four are odd graceful graphs.