一类单圈图的优美性和平衡性

The Gracefulness and Balance of Some Unicyclic Graphs

设L为简单无向图G的一个顶点标号,L称为图G的优美标号,若L满足以下两条:(1)L为G的顶点集V到{0,1,2,…,|E|}的一个单射;(2)由L(′e)=|L(u)-L(v)|(其中e=uv)决定的边标号L′是G的边集E到{1,2,…,|E|}的一个双射.进一步,若存在正整数c,使得对每一个uv∈E(G)满足L(u)≤c<L(v)或L(v)≤c<L(u),则称L为图G的平衡标号,其中c为平衡特征.主要研究一类单圈图的平衡性并给出相应的平衡标号及其特征.

Let L be a vertex label of simple graph G, L is said to be a graceful label of G if L satisfies both the following： （1） Lis an injective mapping from V（G） to {0,1,2,…, ｜E｜ } ; （2） the function L＇obtained by setting L＇ = （e） = ｜ L（u） - L（v） ｜ , where e = uv is an objective function from E（G） to {0,1,2,…, ｜ El} . Furthermore, if there is an integer c for each uv ∈ E（G）, either L（u） ≤ c 〈 L（v） or L（v）≤ c 〈 L（u）, then L is said to be a balanced label of G, and its feature is c. In this paper, we put focus on the balance of some unicyclic graphs, and give their balance label as well as their features.