Graceful Labellings of Various Cyclic Snakes

TL;DR

This paper introduces a new sufficiency condition for graceful labelling of cyclic snakes, successfully applying it to various cases and solving the case for cycles of length 6.

Contribution

It presents a novel sufficiency condition for graceful labelling of cyclic snakes and extends results to various cycle lengths, including a complete solution for length 6.

Findings

Successfully labelled all $kC_{4n}$ snakes for $n=1$ to $6$

Extended labelling results to cyclic snakes with varying cycle lengths

Solved the case for cycles of length 6 completely

Abstract

In this paper, we present a new sufficiency condition to obtain a graceful labelling for every $kC_{4n}$ snake and use this condition to label every such snake for $n=1,2,\ldots,6$. 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 $n=6$.