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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{ADM_2018_25_2_a2,
     author = {Soumya Bhoumik and Sarbari Mitra},
     title = {On $k$-graceful labeling of pendant edge extension of complete bipartite graphs},
     journal = {Algebra and discrete mathematics},
     pages = {188--199},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2018_25_2_a2/}
}
TY  - JOUR
AU  - Soumya Bhoumik
AU  - Sarbari Mitra
TI  - On $k$-graceful labeling of pendant edge extension of complete bipartite graphs
JO  - Algebra and discrete mathematics
PY  - 2018
SP  - 188
EP  - 199
VL  - 25
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2018_25_2_a2/
LA  - en
ID  - ADM_2018_25_2_a2
ER  - 
%0 Journal Article
%A Soumya Bhoumik
%A Sarbari Mitra
%T On $k$-graceful labeling of pendant edge extension of complete bipartite graphs
%J Algebra and discrete mathematics
%D 2018
%P 188-199
%V 25
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2018_25_2_a2/
%G en
%F ADM_2018_25_2_a2
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/

[1] S. Bhoumik, S. Mitra, “Graceful Labeling of Pendant Edge Extension of Complete Bipartite Graph”, International Journal of Mathematical Analysis, 8:58 (2014), 2885–2897 | DOI | MR

[2] J. A. Bondy and U. S. R. Murty, Graph theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008 | DOI | MR | Zbl

[3] D. M. Cvetković, P. Rowlinson, and S. K. Simić, An Introduction to the Theory of Graph Spectra, Cambridge–New York, 2010 | MR | Zbl

[4] J. A. Gallian, “A Dynamic Survey of Graph Labeling”, The Electronics Journal of Combinatorics, 16:3 (2013) | MR

[5] S. W. Golomb, “How to number a graph”, Graph theory and computing, 1972, 23–37 | DOI | MR | Zbl

[6] R. L. Graham, N. J. Sloane, “On additive bases and harmonious graphs”, SIAM Journal on Algebraic Discrete Methods, 1:4 (1980), 382–404 | DOI | MR | Zbl

[7] Y. L. Jirimuta, “On $k$-gracefulness of $r$-Crown $I_r (K_{1,n} ) (n\ge 2,r\ge 2)$ for Complete Bipartite Graph”, Journal of Inner Mongolia University for Nationalities, 2 (2003), 108–110

[8] W. Li, G. Li, and Q. Yan, “Advances in Computer Science, Environment, Ecoinformatics, and Education”, Communications in Computer and Information Science, 214 (2011), 297–301 | DOI | Zbl

[9] Ma Ke-Jie, Graceful Graphs, Beijing University Publishing House, Beijing, 1991

[10] M. Maheo, H. Thuillier, “On $d$-graceful graphs”, Ars Combin., 13 (1982), 181–192 | MR | Zbl

[11] S. Mitra, S. Bhoumik, “On Graceful Labeling Of 1-crown For Complete Bipartite Graph”, International Journal of Computational and Applied Mathematics, 10:1 (2015), 69–75 | MR

[12] A. Rosa, “On certain valuations of the vertices of a graph”, Theory of Graphs, International Symposium (Rome), 1966, 349–355 | MR

[13] G. Sethuraman and A. Elumalai, “On graceful graphs: pendant edge extensions of a family of complete bipartite and complete tripartite graphs”, Indian J. Pure Appl. Math, 32:9 (2001), 1283–1296 | MR | Zbl

[14] P. J. Slater, “On $k$-graceful graphs”, Proc. of the 13th SE Conf. on Combinatorics, Graph Theory and Computing, 1982, 53–57 | MR | Zbl