Let T = (V, E) be a tree where |V| = v and |E| = e. The tree T is called graceful if it is possible to assign the labels {1, 2, 3, .. . , v} to the vertices of T in such a manner that the induced edge labeling - where each edge {i, j} is assigned the label |i - j|, for i, j ∈ {1, 2, 3, . . . , v}, i ≠ j -results in the e edges being labeled by 1, 2, 3, . . . , e.
(a) Prove that every path on n vertices, n ≥ 2, is graceful.
(b) For n ∈ Z+, n ≥ 2, show that K1,n is graceful.
(c) If T = (V, E) is a tree with 4 ≤ |V| ≤ 6, show that T is graceful. (It has been conjectured that every tree is graceful.)