Graceful Labeling of a Tree from Caterpillars

A graph G with vertical bar E(G)vertical bar = q, an injective function f:V(G)-> {0,1,2,...,q} is a beta-labeling of G provided that the values vertical bar f(u)-f(v)vertical bar for the q pairs of adjacent vertices u and v are distinct. A beta-labeling is also known as a graceful labeling. It has been conjectured that every tree is graceful. In this paper, we define graceful labeling for a tree constructed from caterpillars.