The joint sum of graceful trees (Q1324364)

A graph is called graceful, if there is a labeling \(f\) of its vertices with distinct integers so that the induced edge labeling \(f'\) defined by \(f'(u,v) = | f(u) - f(v) |\) assigns each edge a different label. In 1966 A. Rosa conjectured that any tree is graceful. The present paper defines the concept of joint sum of graceful trees: Two trees \(T\) and \(T'\) define the joint sum of \(T\) and \(T'\) as a new tree by connecting a vertex of \(T\) with a vertex of \(T'\). The paper studies the gracefulness of the joint sum of graceful trees and gives necessary and sufficient conditions for a certain family of trees.