# Graceful Labeling in the Context of Duplication of Graph Elements

## Abstract

Let a simple graph G = ((V (G), E(G)) be connected and undirected graph with p vertices and q edges. Let f be a function from the set of vertices to the set of integer {0,1 , 2, 3,…, q} and let f* be a function from the set of edge to the set of positive integer {1, 2, 3,…, q} defined as

f*(uv) = |f(u) − f(v)|

for all u,v∈V (G) and uv∈E(G) . A function f is called graceful labeling of a graph G if f is injective and f* is bijective. The graph G which admits graceful labeling f is called a graceful graph. In this paper, we find graceful labeling for some new graphs results from the duplication of graph elements and then we prove that the new graphs are graceful.

## Article Details

Section
Research Articles

## References

Rosa, A. (1967). On certain valuations of the vertices of a graph. (In Theory of Graphs, International Symposium, Rome, July, 1966), New York: Gordon and Breach.

Golomb, S. W. (1972). How to number a graph. In R. C. Read (Ed.). Graph Theory and Computing. 23–37. New York, U.S.A. Academic Press.

Cattell, R. (2007). Graceful labellings of paths. Discrete Mathematics, 307(24), 3161–3176.

Kaneria, V. J., Makadia, H. M., & Jariya, M. M. (2014). Graceful labeling for cycle of graph. International Journal of Mathematics Research, 6(2), 173–178.

Boxwala, S. A., & Vashishta, P. (2015). Some new families of graceful graphs. Electronic Notes in Discrete Mathematics, 48, 127–133.

Koh, K. M., Phoon, L. Y., & Soh, K. W. (2015). The gracefulness of the join of graphs (II). AKCE International Journal of Graphs and Combinatorics, 12(2-3) 180–185.

Hegde, S. M., & Kumudakshi. (2016). Forther results on graceful directed graphs. Electronic Notes in Discrete Mathematics, 53, 149–154.

Khatun, S., & Nayeem, A. (2017). Graceful labeling of some zero divisor graphs. Electronic Notes in Discrete Mathematics, 63, 189–196.

Ragukumar, P., & Sethuraman, G. (2018). Binomial trees are graceful, AKCE International Journal of Graphs and Combinatorics, In Press.

Wang, T. M., & Zhang, G. H. (2018). On edge–graceful labeling and deficiency for regular graphs. AKCE International Journal of Graphs and Combinatorics, 15 (1), 105–111.