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Abstract
In this paper, we introduce new results on k-prime labeling. First, we discuss the kprime labeling of cycles Cn for some values of k and n . Also we give the k-prime labeling of combs Pn K1and some case of the crowns Cn K1 . Second, we show that all wheels W 2n> 1 are not k-prime for every positive integers k and n while W2n (n 2) is not k-prime for every even positive integers k . Finally, we give the k-prime labeling of the helm Hn (n 3) for 2 k 11 and we show that if k 2(n 1) and k 2n are twin primes where n 3 and k 1, then Hn is k -prime.
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References
T. Deretsky, S. M. Lee, and J. Mitchem, On vertex prime labelings of graphs, in Graph Theory, Combinatorics and Applications Vol. 1, J. Alavi, G. Chartrand, O. Oellerman, and A. Schwenk, eds., Proceedings 6th International Conference Theory and Applications of Graphs (Wiley, New York, 1991) 359-369.
N. Diefenderfer, M. Hastings, L. N. Heath, H. Prawzinsky, B. Preston, E. White, and A. Whittemore, Prime vertex labelings of families of unicyclic graphs, Rose-Hulman Undergrad. Math J., 16 (1) (2015) 253-269.
H. L. Fu and K. C. Huang, On prime labelings , Discrete Mathematics, 127(1994) 181-186.
J. A. Gallian, A dynamic survey of graph labeling, Elect. J. Combin., 18 edition , #DS6 (2015) 1-389.
A. Kanetkar, Prime labeling of grids, AKCE J. Graphs Combin., 6 (2009) 135-142.
S. M. Lee, I. Wui and J. Yeh, On the amalgamation of prime graphs, Bull. Malaysian Math. Soc. (Second Series), 11 (1988) 59-67.
A. G. Michael and M. Z. Youssef, On prime self-complementary graphs, J. Disc. Math. Sci. and Cryptography, 17 (3) (2014) 239-256.
M. A. Seoud, A. El Sonbaty, and A. E. A. Mahran, On prime graphs, Ars Combin., 104 (2012) 241-260.
M. A. Seoud A. El Sonbaty, and A. E. A. Mahran, Primality of some graphs, Ars Combin., 112 (2013) 459-469.
M. A. Seoud and M. Z. Youssef, On prime labelings of graphs, Congr. Numer., 141 (1999) 203-215.
M. Sundaram, R. Ponraj, and S. Somasundaram, On a prime labeling conjecture, Ars Combin., 80 (2006) 205-209.
A. Tout, A. N. Dabboucy and K. Howalla, Prime labeling of graphs, Nat. Acad.Sci. Letters, 11, (1982) 365–368.
S. K. Vaidya and U. M. Prajapati, Some results on prime and k-prime labeling, J. Math. Research, 3(1) (2011) 66-75.
M. Z. Youssef and E.A. El Sakhawi, Some properties of prime graphs, Ars Combin., 84 (2007) 129-140.