Every snowflake graph is conservative exactly when its edge count is 0 or 3 modulo 4, and large two-nested-cycle graphs admit matching graceful or near-graceful labelings.
Graceful Labellings of Various Cyclic Snakes
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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 $n=6$.
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Gracefulness of two nested cycles: a first approach
Every snowflake graph is conservative exactly when its edge count is 0 or 3 modulo 4, and large two-nested-cycle graphs admit matching graceful or near-graceful labelings.