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) C4- snake and (2, k) C4- snake are odd graceful. We prove that the linear cyclic snakes (1, k) C6- snake and (2, k) C6- snake are odd graceful. We also prove that the linear cyclic snakes (1, k) C8- snake and (2, k) C8- snake are odd graceful. We generalize the above results "the linear cyclic snakes (m, k) C4- snake, (m, k) C6-snake and (m, k) C8-snake are odd graceful ". Finally, we introduce a new conjecture" All the linear cyclic snakes (m, k) Cn-snakes are odd graceful if n is even)".