# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021029
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## On ${L}(2,1)$-labelings of some products of oriented cycles

 1 Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, 1053 Budapest, Hungary 2 Central European University, Department of Mathematics and its Applications, Nádor u. 9, 1051 Budapest, Hungary

* Corresponding author: Lucas Colucci

Received  December 2019 Revised  July 2021 Early access November 2021

Fund Project: 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

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 $m$, $n \geq 48$.

Citation: Lucas Colucci, Ervin Győri. On ${L}(2,1)$-labelings of some products of oriented cycles. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021029
##### References:
 [1] T. Calamoneri, The ${L}(h, k)$-labelling problem: An updated survey and annotated bibliography, 2014, Available on http://www.dsi.uniroma1.it/calamo/PDF-FILES/survey.pdf. Google Scholar [2] G. J. Chang and S.-C. Liaw, The ${L}(2, 1)$-labeling problem on ditrees, Ars Combin., 66 (2003), 23-31.   Google Scholar [3] W. K. Hale, Frequency assignment: Theory and applications, Proceedings of the IEEE, 68 (1980), 1497-1514.  doi: 10.1109/PROC.1980.11899.  Google Scholar [4] Z. Shao, H. 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.  Google Scholar [5] J. J. Sylvester, et al, Mathematical questions with their solutions, Educational times 41, 21 (1884), 6pp. Google Scholar [6] K. -C. Yeh, Labeling Graphs with a Condition at Distance Two, PhD thesis, University of South Carolina, 1990. Google Scholar [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.  Google Scholar

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##### References:
 [1] T. Calamoneri, The ${L}(h, k)$-labelling problem: An updated survey and annotated bibliography, 2014, Available on http://www.dsi.uniroma1.it/calamo/PDF-FILES/survey.pdf. Google Scholar [2] G. J. Chang and S.-C. Liaw, The ${L}(2, 1)$-labeling problem on ditrees, Ars Combin., 66 (2003), 23-31.   Google Scholar [3] W. K. Hale, Frequency assignment: Theory and applications, Proceedings of the IEEE, 68 (1980), 1497-1514.  doi: 10.1109/PROC.1980.11899.  Google Scholar [4] Z. Shao, H. 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.  Google Scholar [5] J. J. Sylvester, et al, Mathematical questions with their solutions, Educational times 41, 21 (1884), 6pp. Google Scholar [6] K. -C. Yeh, Labeling Graphs with a Condition at Distance Two, PhD thesis, University of South Carolina, 1990. Google Scholar [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.  Google Scholar
The Cartesian product of $\overrightarrow {{P_3}}$ and $\overrightarrow {{P_4}}$
The strong product of $\overrightarrow {{P_3}}$ and $\overrightarrow {{P_4}}$
A $\overrightarrow{P_4} \boxtimes \overrightarrow{P_4}$ subgraph of $\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}$
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