Korean Journal of Mathematics

  • 제25권2호
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  • Pages.279-301
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  • 2017
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  • 1976-8605(pISSN)
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  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
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L(3, 2, 1)-LABELING FOR CYLINDRICAL GRID: THE CARTESIAN PRODUCT OF A PATH AND A CYCLE

  • 투고 : 2017.01.20
  • 심사 : 2017.06.21
  • 발행 : 2017.06.30
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⟨ 이전 논문 다음 논문 ⟩

초록

An L(3, 2, 1)-labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that ${\mid}f(u)-f({\upsilon}){\mid}{\geq}4-k$ when $dist(u,{\upsilon})=k{\leq}3$. The L(3, 2, 1)-labeling number, denoted by ${\lambda}_{3,2,1}(G)$, for G is the smallest number N such that there is an L(3, 2, 1)-labeling for G with span N. In this paper, we compute the L(3, 2, 1)-labeling number ${\lambda}_{3,2,1}(G)$ when G is a cylindrical grid, which is the cartesian product $P_m{\Box}C_n$ of the path and the cycle, when $m{\geq}4$ and $n{\geq}138$. Especially when n is a multiple of 4, or m = 4 and n is a multiple of 6, then we have ${\lambda}_{3,2,1}(G)=11$. Otherwise ${\lambda}_{3,2,1}(G)=12$.

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참고문헌

  1. A. A. Bertossi and M. C. Pinotti, Channel assignment with separation for inter-ference avoidence in weierless networks, IEEE Trans. Parall. Distrib. Syst. 14 (2003), 222-235. https://doi.org/10.1109/TPDS.2003.1189581
  2. A. A. Bertossi and M. C. Pinotti, Approximate L(${\delta}_1,\;{\delta}_2,\;...\;,\;{\delta}_t$)-coloring of trees and interval graphs, Networks 49 (3) (2007), 204-216. https://doi.org/10.1002/net.20154
  3. A. A. Bertossi, M. C. Pinotti and R. Tan, Efficient use of Radio spectrum in weierless networks with channel separation between close stations, dial M for mobility: International ACM workshop, Discrete algorithms and methods for mobile computing, (2000).
  4. T. Calamoneri, The L(h; k)-labeling problem: A survey and annotated bibliogra-phy Comp. J. 49 (2006), 585-608. https://doi.org/10.1093/comjnl/bxl018
  5. T. Calamoneri, The L(h; k)-Labelling Problem: An Updated Survey and Anno-tated Bibliography, Comp. J. 54 (2011), 1344-1371. https://doi.org/10.1093/comjnl/bxr037
  6. T. Calamoneri, Optimal L(${\delta}_1,\;{\delta}_2,\;1$)- labelling of eight-regular grids, Inform. Pro-cess. Lett., 113 (2013), 361-364. https://doi.org/10.1016/j.ipl.2013.03.003
  7. G. J. Chang and D. Kuo, The L(2; 1)-labeling problem on graphs, SIAM J Dis-crete Math., 9, 309-316 (1996). https://doi.org/10.1137/S0895480193245339
  8. M. L. Chia, D. Kuo, H. Liao, C. H. Yang and R. K. Yeh, L(3; 2; 1) labeling of graphs, Taiwan. J. Math. 15 (2011), 2439-2457. https://doi.org/10.11650/twjm/1500406480
  9. J. R. Griggs and R. K. Yeh, Labeling graphs with a condition at distance two, SIAM J. Discrete Math. 5 (1992), 586-595. https://doi.org/10.1137/0405048
  10. W. K. Hale, Frequency assignment: theory and application, Proc. IEEE 68 (1980), 1497-1514. https://doi.org/10.1109/PROC.1980.11899
  11. B. M. Kim, W. Hwang and B. C. Song, L(3; 2; 1)-labeling for the product of a complete graph and a cycle, Taiwan. J. Math., 19 (2015), 819-848. https://doi.org/10.11650/tjm.19.2015.4449
  12. B. M. Kim, B. C. Song and W. Hwang, Distance three labelings for direct products of three complete graphs, Taiwan. J. Math. 17 (2013), 207-219. https://doi.org/10.11650/tjm.17.2013.1909
  13. B. M. Kim, B. C. Song and W. Hwang, Distance three labellings for $K_n{\times}K_2$, Int. J. Comput. Math., 90 (5) (2013), 906-911. https://doi.org/10.1080/00207160.2012.748896
  14. J. Liu and Z. Shao, The L(3; 2; 1)-labeling problem on graphs, Math. Appl. 17 (2004), 596-602.
  15. Z. Shao and A. Vesel, Integer linear programming model and satisfiability test reduction for distance constrained labellings of graphs: the case of L(3; 2; 1) la-belling for products of paths and cycles, IET Commun. 7 (2013), 715-720. https://doi.org/10.1049/iet-com.2012.0568
  16. Z. Shao and A. Vesel, L(3,2,1) -labeling of triangular and toroidal grids, Cent. Eur. J. Oper. Res. 23 (2015), 659-673. https://doi.org/10.1007/s10100-014-0365-4
  17. R. K. Yeh, A survey on labeling graphs with a condition two, Disc. Math., 306 (2006)., 1217-123 https://doi.org/10.1016/j.disc.2005.11.029
  18. S. Zhou, A distance-labelling problem for hypercubes, Discrete Appl. Math. 156 (15) (2008), 2846-2854. https://doi.org/10.1016/j.dam.2007.11.018