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On $ {L}(2,1) $-labelings of some products of oriented cycles

  • * Corresponding author: Lucas Colucci

    * Corresponding author: Lucas Colucci 
The first author is partially supported by the National Research, Development and Innovation, NKFIH grant K 116769. The second author is partially supported by the National Research, Development and Innovation, NKFIH grants K 116769 and SNN 117879
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  • We refine two results of Jiang, Shao and Vesel on the $ L(2,1) $-labeling number $ \lambda $ of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of $ \lambda(\overrightarrow{C_m} \square \overrightarrow{C_n}) $ for $ m $, $ n \geq 40 $; in the case of strong product, we either compute the exact value or establish a gap of size one for $ \lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}) $ for $ m $, $ n \geq 48 $.

    Mathematics Subject Classification: Primary: 05C15, 05C78; Secondary: 05C20.


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  • Figure 1.  The Cartesian product of $\overrightarrow {{P_3}} $ and $\overrightarrow {{P_4}} $

    Figure 2.  The strong product of $\overrightarrow {{P_3}} $ and $\overrightarrow {{P_4}} $

    Figure 3.  A $ \overrightarrow{P_4} \boxtimes \overrightarrow{P_4} $ subgraph of $ \overrightarrow{C_m} \boxtimes \overrightarrow{C_n} $

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    [2] G. J. Chang and S.-C. Liaw, The ${L}(2, 1) $-labeling problem on ditrees, Ars Combin., 66 (2003), 23-31. 
    [3] W. K. Hale, Frequency assignment: Theory and applications, Proceedings of the IEEE, 68 (1980), 1497-1514.  doi: 10.1109/PROC.1980.11899.
    [4] Z. ShaoH. Jiang and A. Vesel, ${L}(2, 1) $-labeling of the cartesian and strong product of two directed cycles, Mathematical Foundations of Computing, 1 (2018), 49-61.  doi: 10.3934/mfc.2018003.
    [5] J. J. Sylvester, et al, Mathematical questions with their solutions, Educational times 41, 21 (1884), 6pp.
    [6] K. -C. Yeh, Labeling Graphs with a Condition at Distance Two, PhD thesis, University of South Carolina, 1990.
    [7] R. K. Yeh, A survey on labeling graphs with a condition at distance two, Discrete Math., 306 (2006), 1217-1231.  doi: 10.1016/j.disc.2005.11.029.
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