# Graceful Trees and Paths

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A (p,q) graph G is called graceful if it is possible to label the vertices of G with distinct

elements from the set {0,1,...,q} in such a way that the induced edge labeling, which

assigns the integer |i - j| to the edge ij, assigns the labels 1,2,...,q to the q edges of G.

The graceful tree conjecture states that every tree is graceful.

a. Prove that every path is graceful.

b. Prove that every star K_(1,n) is graceful.

c. Show that every tree of order 6 is graceful.

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Graceful trees and paths are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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============================

A (p,q) graph G is called graceful if it is possible to label the vertices of G with distinct

elements from the set {0,1,...,p} in such a way that the induced edge labeling, which

assigns the integer |i - j| to the edge ij, assigns the labels 1,2,...,q to the q edges of G.

The ...

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