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.