Geodesic
On $k$-graceful labeling of pendant edge extension of complete bipartite graphs
Algebra and discrete mathematics, Tome 25 (2018) no. 2, pp. 188-199

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For any two positive integers $m,n$, we denote the graph $K_{m,n}\odot K_1$ by $G$. Ma Ke-Jie proposed a conjecture [9] that pendant edge extension of a complete bipartite graph is a $k$-graceful graph for $k \ge 2$. In this paper we prove his conjecture for $n\le m n^2+\lfloor\frac{k}{n}\rfloor+ r$.
Keywords: $k$-graceful labeling, complete bipartite graph, corona, $1$-crown. 
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     author = {Soumya Bhoumik and Sarbari Mitra},
     title = {On $k$-graceful labeling of pendant edge extension of complete bipartite graphs},
     journal = {Algebra and discrete mathematics},
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     url = {http://geodesic.mathdoc.fr/item/ADM_2018_25_2_a2/}
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Soumya Bhoumik; Sarbari Mitra. On $k$-graceful labeling of pendant edge extension of complete bipartite graphs. Algebra and discrete mathematics, Tome 25 (2018) no. 2, pp. 188-199. http://geodesic.mathdoc.fr/item/ADM_2018_25_2_a2/