On the gracefulness of the digraph P→a,b

Abstract

A digraph D(V, E) is said to be graceful if there exists an injection f: V(G) → {0,1, • • •, |E|}, such that the induced function f': E(G) →{1,2, • • •, which is defined by f'(u, v) = (f(v) - f(u)) (mod |E| + 1) for every directed edge (u,v) is a bijection. Here, f is called a graceful labeling of D(V,E), and f' is called the induced edge's graceful labeling of D. In this paper we discuss the gracefulness of the digraph P→_a,b, and prove P→_a,b is graceful if and only if both a and b are odd.