於西元1967年,Rosa為首位使用圖形標號作為圖形分割工具的學者.從那時起,就有相當多不同的標號種類被研究出來.近來,J.A. Gallian所著論述,將標號問題的研究做全面性整理.
Rosa 首先介紹Beta標號和Alpha標號.特別是後者,對於分割一圖至循環同構子圖提供了一個重要的工具.因此,知道許多圖形具有Alpha標號是令人感興趣地.
令圖形G擁有q個邊(edges),有一對映函數(injective function)f:V(G)---> {0,1,2,..., q},是將圖形G的每一節點(vertices)從0至q加以編號,使得每對相鄰的節點uv,|f(u)-f(v)|的值皆不同(distinct),則我們稱此函數f為圖形G的一個Beta標號(Beta-labeling).一個Beta標號又稱作是一個完美標號(Graceful labeling).而Alpha標號(Alpha-labeling)為完美標號再增加一性質,即存在一整數lambda,使得對於每邊uv,滿足f(u)<= lambda
Let G be a graph with q edges, we call a function f a β-labeling of G if f is an injective
function from the vertices of G to{0,1,2, . . . , q} such that all values |f(u)−f(v)| for the q pairs of adjacent vertices u and v are distinct. A β-labeling is also known as a graceful labeling. An α-labeling is a graceful labeling with additional property that there is an integer λ such that for each edge uv either f(u)<=λ