Graceful Labellings of Various Cyclic Snakes

In this paper, we present a new sufficiency condition to obtain a graceful labelling for every $k{C}_{4n}$ snake and use this condition to label every such snake for $n=1,2,\dots ,6$. Then, we extend this result to cyclic snakes where the cycles lengths vary. Also, we obtain new results on the (near) graceful labelling of cyclic snakes based on cycles of lengths $n=6,10,14$, completely solving the case $n=6$.