On the gracefulness of the digraphs n - C→19 for even n

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), and 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^→₁₉ is graceful for even n.