ON THE GRACEFULNESS OF THE DIGRAPHS n ¡ -Cm FOR m ODD

A digraph D(V;E) is said to be graceful if there exists aninjection f : V (G) ! f0; 1; ¢ ¢ ¢ ; jEjg such that the induced function f0 :E(G) ! f1; 2; ¢ ¢ ¢ ; jEjg which is de¯ned by f0 (u; v) = [f(v)¡f(u)] (mod jEj+1) for every directed edge (u; v) is a bijection. Here, f is called a graceful labeling (graceful numbering) of D(V;E), while f0 is called the induced edgeʹs graceful labeling of D. In this paper we discuss the gracefulness of the digraph n- Cm and prove that n-Cm is a graceful digraph for m = 5; 7; 9; 11; 13 and even n.