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Bound state solutions for fractional Schrödinger-Poisson systems
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 |
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 $.
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. |
[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. 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. |
[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. |
show all references
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. |
[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. 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. |
[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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