Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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THE λ-NUMBER OF THE CARTESIAN PRODUCT OF A COMPLETE GRAPH AND A CYCLE

  • Kim, Byeong Moon (Department of Mathematics Gangneung-Wonju National University) ;
  • Song, Byung Chul (Department of Mathematics Gangneung-Wonju National University) ;
  • Rho, Yoomi (Department of Mathematics University of Incheon)
  • Received : 2013.03.13
  • Accepted : 2013.05.20
  • Published : 2013.06.30
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Abstract

An $L(j,k)$-labeling of a graph G is a vertex labeling such that the difference of the labels of any adjacent vertices is at least $j$ and that of any vertices of distance two is at least $k$ for given $j$ and $k$. The minimum span of all L(2, 1)-labelings of G is called the ${\lambda}$-number of G and is denoted by ${\lambda}(G)$. In this paper, we find a lower bound of the ${\lambda}$-number of the Cartesian product $K_m{\Box}C_n$ of the complete graph $K_m$ of order $m$ and the cycle $C_n$ of order $n$. In fact, we show that when $n{\geq}3$, ${\lambda}(K_4{\Box}C_n){\geq}7$ and the equality holds if and only if n is a multiple of 8. Moreover when $m{\geq}5$, ${\lambda}(K_m{\Box}C_n){\geq}2m-1$ and the equality holds if and only if $n$ is even.

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Acknowledgement

Supported by : University of Incheon

References

  1. S.S. Adams, J. Cass and D.S. Troxell, An extension of the channel-assignment problem: L(2, 1)-labelings of generalized Petersen graphs, IEEE Trans. Circ. Sys. I-Regular Papers, 53 (2006), 1101-1107. https://doi.org/10.1109/TCSI.2005.862184
  2. T. Calamoneri, The L(h, k)-labeling problem: A survey and annotated biblography, Computer Journal, 49 (2006), 585-608. https://doi.org/10.1093/comjnl/bxl018
  3. T. Calamoneri, The L(h, k)-labeling problem: An updated survey and anno-tated biblography, Computer Journal, 54 (2011), 1344-1371. https://doi.org/10.1093/comjnl/bxr037
  4. R. Erman, S. Jurecic, D. Kral, K. Stopar and N. Stopar, Optimal real number graph labellings of a subfamily of Kneser graphs, (English summary), SIAM J. Disc. Math. 23 (2009), 1372-1381. https://doi.org/10.1137/060672923
  5. J.R. Griggs and X.T. Jin, Recent progress in mathematics and engineering on optimal graph labelings with distance conditions, J. of Comb. Optim. 14 (2007), 249-257. https://doi.org/10.1007/s10878-007-9055-7
  6. J.R. Griggs and R.K. Yeh, Labeling graphs with acondition at distance two, SIAM J. Disc. Math 5 (1992), 586-595. https://doi.org/10.1137/0405048
  7. J.P. Georges, D.W. Mauro and M. I. Stein, Labeling products of complete graphs with a condition at a distance two, SIAM J. Disc. Math. 14 (2000), 2-35.
  8. W.K. Hale, Frequency assignment: theory and application, Proc. IEEE 68 (1980), 1497-1514. https://doi.org/10.1109/PROC.1980.11899
  9. E. Haque and P. K. Jha, L(j, k)-labelings of Kronecker products of complete graphs, IEEE Trans. on Cir. and Sys. II 55 (2008), 70-73. https://doi.org/10.1109/TCSII.2007.908918
  10. F. Havet, B. Reed and J. Sereni, Griggs and Yeh's Conjecture and L(p, 1)-labelings, SIAM J. Disc. Math. 26 (2012), 145-168. https://doi.org/10.1137/090763998
  11. P.K. Jha, S. Klavzar and A. Vesel, Optimal L(d, 1)-labelings of certain direct products of cycles and Cartesian product of cycles, Disc. Appl. Math. 152 (2005), 257-265. https://doi.org/10.1016/j.dam.2005.04.007
  12. J. Kang, L(2, 1)-labeling of hamiltonian graphs with maximum degree 3, SIAM J. Disc. Math. 22 (2008), 213-230. https://doi.org/10.1137/050632609
  13. S. Klavzar and S. Spacapan, The ${\Delta}^2$-conjecture for L(2, 1)-labelings is true for directed and strong products of graphs, IEEE Trans. Circuits Syst. II 53(2006), 274-277.
  14. B. Kim, B. Song and Y. Rho, Labeling products of complete graphs with a condition at a distance two, International J. Computer Mathematics, in press.
  15. W. Lin and P.C.B. Lam, Distance two labeling and direct products of graphs, Disc. Math. 308 (2008), 3805-3815. https://doi.org/10.1016/j.disc.2007.06.046
  16. P. Lam, W. Lin and J. Wu, L(j, k)-labelings for the products of complete graphs, J. of Comb. Optim. 14 (2007), 219-227. https://doi.org/10.1007/s10878-007-9057-5
  17. C. Schwartz and D. Troxell, L(2, 1)-labelings of Cartesian product of two cycles, Disc. Appl. Math. 154 (2006), 1522-1540. https://doi.org/10.1016/j.dam.2005.12.006
  18. Z.D. Shao and R.K. Yeh, The L(2, 1)-labeling and operations of graphs, IEEE Trans. Circ. Sys. I-Regular Papers 52 (2005), 668-671. https://doi.org/10.1109/TCSI.2004.840484
  19. M.A. Whittlesey, J.P. Georges and D.W. Mauro, On the ${\lambda}$-number of Qn and related graphs, SIAM J. Disc. Math. 8 (1995), 499-506. https://doi.org/10.1137/S0895480192242821

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