A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from {0, 1, ..., q} to each vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v) where g(u, v) = (g(v)− g(u))(mod q + 1). If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. In this paper, we prove a general result on graceful digra...

A function f is called a graceful labeling of a graph G with m edges if f is an injective function from V (G) to {0, 1, 2, · · · ,m} such that when every edge uv is assigned the edge label |f(u)− f(v)|, then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. The popular Graceful Tree Conjecture states that every tree is graceful. The Gra...

The objective of this paper is to present a new class of odd graceful graphs. In particular, we show that the linear cyclic snakes (1, k) C4snake and (2, k) C4snake are odd graceful. We prove that the linear cyclic snakes (1, k) C6snake and (2, k) C6snake are odd graceful. We also prove that the linear cyclic snakes (1, k) C8snake and (2, k) C8snake are odd graceful. We generalize the above res...