The aim of this paper is to present a new class of graceful graphs. In particular, we show that linear cyclic snakes (1, k) C4- snake and (2, k) C4- snake are graceful. We also prove that linear cyclic snakes (1, k) C8- snake and (2, k) C8- snake are graceful. We generalize the above results " the linear cyclic snakes (m, k) C4- snake and (m, k) C8- snake are graceful ". Finally, we introduce a new conjecture" All the linear cyclic snakes (m, k) Cn-snakes are graceful if n 0 (mod 4 ) or n 3 (mod 4 )".