On the gracefulness of the digraphs n - C→m

Abstract

A digraph D(V, E) is said to be graceful if there exists an injection f : V(D) -→ {0, 1, ⋯ , |E|} such that the induced function f′ : E(D) -→ {1,2, ⋯ ,|E|} 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(graceful numbering) of digraph D(V, E), while f′ is called the induced edge's graceful labeling of digraph D(V, E). In this paper, we discuss the gracefulness of the digraph n - C→_m and prove the digraph n - C→₁₅ is graceful for even n.