    February  2018, 1(1): 49-61. doi: 10.3934/mfc.2018003

## L(2, 1)-labeling of the Cartesian and strong product of two directed cycles

 1 Research Institute of Intelligence Software, Guangzhou University, Guangzhou 510006, China 2 School of Information Science and Engineering, Chengdu University, Chengdu 610106, China 3 Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia

Received  October 2017 Revised  December 2017 Published  February 2018

Fund Project: Supported by the National Key Research and Development Program under grant 2017YFB0802300, the National Natural Science Foundation of China under the grant No. 61309015 and by the Ministry of Science of Slovenia under the grant 0101-P-297, and Applied Basic Research (Key Project) of Sichuan Province under grant 2017JY0095.

The frequency assignment problem (FAP) is the assignment of frequencies to television and radio transmitters subject to restrictions imposed by the distance between transmitters. One of the graph theoretical models of FAP which is well elaborated is the concept of distance constrained labeling of graphs. Let $G = (V, E)$ be a graph. For two vertices $u$ and $v$ of $G$, we denote $d(u, v)$ the distance between $u$ and $v$. An $L(2, 1)$-labeling for $G$ is a function $f: V → \{0, 1, ···\}$ such that $|f(u)-f(v)| ≥ 1$ if $d(u, v) = 2$ and $|f(u)-f(v)| ≥ 2$ if $d(u, v) = 1$. The span of $f$ is the difference between the largest and the smallest number of $f(V)$. The $λ$-number for $G$, denoted by $λ(G)$, is the minimum span over all $L(2, 1)$-labelings of $G$. In this paper, we study the $λ$-number of the Cartesian and strong product of two directed cycles. We show that for $m, n ≥ 4$ the $λ$-number of $\overrightarrow{C_m} \Box \overrightarrow{C_n}$ is between 4 and 5. We also establish the $λ$-number of $\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$ for $m ≤ 10$ and prove that the $λ$-number of the strong product of cycles $\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$ is between 6 and 8 for $m, n ≥ 48$.

Citation: Zehui Shao, Huiqin Jiang, Aleksander Vesel. L(2, 1)-labeling of the Cartesian and strong product of two directed cycles. Mathematical Foundations of Computing, 2018, 1 (1) : 49-61. doi: 10.3934/mfc.2018003
##### References:
  K. I. Aardal, S. P. M. van Hoesel, A. M. C. A. Koster, C. Mannino and A. Sassano, Models and solution techniques for frequency assignment problems, Ann. Oper. Res., 153 (2007), 79-129.  doi: 10.1007/s10479-007-0178-0.  Google Scholar  H. L. Bodlaender, T. Kloks, R. B. Tan and J. van Leeuwen, Approximations for λ-coloring of graphs, The Computer Journal, 47 (2004), 193-204.   Google Scholar  M. Bouznif, J. Moncel and M. Preissmann, Generic algorithms for some decision problems on fasciagraphs and rotagraphs, Discrete Math., 312 (2012), 2707-2719.  doi: 10.1016/j.disc.2012.02.013.  Google Scholar  T. Calamoneri and B. Sinaimeri, L(2, 1)-labeling of oriented planar graphs, Discrete Appl. Math., 161 (2013), 1719-1725.  doi: 10.1016/j.dam.2012.07.009.  Google Scholar  T. Calamoneri, The L(2, 1)-labeling problem on oriented regular grids, The Computer Journal, 54 (2011), 1869-1875.   Google Scholar  T. Calamoneri, The L(h, k)-labelling problem: An updated survey and annotated bibliography, The Computer Journal, 54 (2011), 1344-1371.  doi: 10.1093/comjnl/bxr037. Google Scholar  G. J. Chang and D. Kuo, The L(2, 1)-labeling problem on graphs, SIAM J. Discrete Math., 9 (1996), 309-316.  doi: 10.1137/S0895480193245339.  Google Scholar  G. J. Chang and S. Liaw, The L(2, 1)-labeling problem on ditrees, Ars Combin., 66 (2003), 23-31. Google Scholar  G. J. Chang, J. Chen, D. Kuo and S. Liaw, Distance-two labelings of digraphs, Discrete Appl. Math., 155 (2007), 1007-1013.  doi: 10.1016/j.dam.2006.11.001.  Google Scholar  Y. Chen, M. Chia and D. Kuo, L(p, q)-labeling of digraphs, Discrete Appl. Math., 157 (2009), 1750-1759.  doi: 10.1016/j.dam.2008.12.007.  Google Scholar  J. Fiala, T. Kloks and J. Kratochvíl, Fixed-parameter complexity of λ-labelings, Discrete Appl. Math., 113 (2001), 59-72.  doi: 10.1016/S0166-218X(00)00387-5.  Google Scholar  J. Fiala, P. A. Golovach and J. Kratochvíl, Distance constrained labelings of graphs of bounded treewidth, Proc. 32th ICALP, 3580 (2005), 360-372. Google Scholar  D. Goncalves, On the L(p, 1)-labelling of graphs, Discrete Math., 308 (2008), 1405-1414.  doi: 10.1016/j.disc.2007.07.075.  Google Scholar  J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance 2, SIAM J. Discrete Math., 5 (1992), 586-595.  doi: 10.1137/0405048.  Google Scholar  W. K. Hale, Frequency assignment: Theory and applications, Proc. IEEE, 68 (1980), 1497-1514.  doi: 10.1109/PROC.1980.11899. Google Scholar  R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, 2nd edition, CRC Press, Boca Raton, 2011. Google Scholar  P. K. Jha, S. Klavžar and A. Vesel, L(2, 1)-labeling of direct product of paths and cycles, Discrete. Appl. Math., 145 (2005), 317-325.  doi: 10.1016/j.dam.2004.01.019.  Google Scholar  S. Klavžar and A. Vesel, Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of (2, 1)-colorings and independence numbers, Discrete Appl. Math., 129 (2003), 449-460.  doi: 10.1016/S0166-218X(02)00597-8.  Google Scholar  D. Korže and A. Vesel, L(2, 1)-labeling of strong products of cycles, Inf. Process. Lett., 94 (2005), 183-190.  doi: 10.1016/j.ipl.2005.01.007.  Google Scholar  D. Král and R. Škrekovski, A theorem about channel assignment problem, SIAM J. Discrete Math., 16 (2003), 426-437.  doi: 10.1137/S0895480101399449.  Google Scholar  J. Kratochvíl, D. Kratsch and M. Liedloff, Exact algorithms for L(2, 1)-labeling of graphs, Proc. 32nd MFCS, 4708 (2007), 513-524. Google Scholar  M. Liang, X. Xu, J. Liang and Z. Shao, Upper bounds on the connection probability for 2-D meshes and tori, J. Parallel and Distrib. Comput., 72 (2012), 185-194.  doi: 10.1016/j.jpdc.2011.11.006. Google Scholar  S. Sen, 2-dipath and oriented L(2, 1)-labelings of some families of oriented planar graphs, Electronic Notes in Discrete Math., 38 (2011), 771-776.  doi: 10.1016/j.endm.2011.10.029. Google Scholar  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)-labelling for products of paths and cycles, IET Communications, 7 (2013), 715-720.   Google Scholar  J. J. Sylvester, Mathematical questions with their solutions, Educational Times, 41 (1884), 171-178.   Google Scholar  M. El-Zahar, S. Khamis and K. Nazzal, On the Domination number of the Cartesian product of the cycle of length n and any graph, Discrete Appl. Math., 155 (2007), 515-522.  doi: 10.1016/j.dam.2006.07.003.  Google Scholar  X. Zhang and J. Qian, L(p, q)-labeling and integer flow on planar graphs, The Computer Journal, 56 (2013), 785-792.   Google Scholar

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##### References:
  K. I. Aardal, S. P. M. van Hoesel, A. M. C. A. Koster, C. Mannino and A. Sassano, Models and solution techniques for frequency assignment problems, Ann. Oper. Res., 153 (2007), 79-129.  doi: 10.1007/s10479-007-0178-0.  Google Scholar  H. L. Bodlaender, T. Kloks, R. B. Tan and J. van Leeuwen, Approximations for λ-coloring of graphs, The Computer Journal, 47 (2004), 193-204.   Google Scholar  M. Bouznif, J. Moncel and M. Preissmann, Generic algorithms for some decision problems on fasciagraphs and rotagraphs, Discrete Math., 312 (2012), 2707-2719.  doi: 10.1016/j.disc.2012.02.013.  Google Scholar  T. Calamoneri and B. Sinaimeri, L(2, 1)-labeling of oriented planar graphs, Discrete Appl. Math., 161 (2013), 1719-1725.  doi: 10.1016/j.dam.2012.07.009.  Google Scholar  T. Calamoneri, The L(2, 1)-labeling problem on oriented regular grids, The Computer Journal, 54 (2011), 1869-1875.   Google Scholar  T. Calamoneri, The L(h, k)-labelling problem: An updated survey and annotated bibliography, The Computer Journal, 54 (2011), 1344-1371.  doi: 10.1093/comjnl/bxr037. Google Scholar  G. J. Chang and D. Kuo, The L(2, 1)-labeling problem on graphs, SIAM J. Discrete Math., 9 (1996), 309-316.  doi: 10.1137/S0895480193245339.  Google Scholar  G. J. Chang and S. Liaw, The L(2, 1)-labeling problem on ditrees, Ars Combin., 66 (2003), 23-31. Google Scholar  G. J. Chang, J. Chen, D. Kuo and S. Liaw, Distance-two labelings of digraphs, Discrete Appl. Math., 155 (2007), 1007-1013.  doi: 10.1016/j.dam.2006.11.001.  Google Scholar  Y. Chen, M. Chia and D. Kuo, L(p, q)-labeling of digraphs, Discrete Appl. Math., 157 (2009), 1750-1759.  doi: 10.1016/j.dam.2008.12.007.  Google Scholar  J. Fiala, T. Kloks and J. Kratochvíl, Fixed-parameter complexity of λ-labelings, Discrete Appl. Math., 113 (2001), 59-72.  doi: 10.1016/S0166-218X(00)00387-5.  Google Scholar  J. Fiala, P. A. Golovach and J. Kratochvíl, Distance constrained labelings of graphs of bounded treewidth, Proc. 32th ICALP, 3580 (2005), 360-372. Google Scholar  D. Goncalves, On the L(p, 1)-labelling of graphs, Discrete Math., 308 (2008), 1405-1414.  doi: 10.1016/j.disc.2007.07.075.  Google Scholar  J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance 2, SIAM J. Discrete Math., 5 (1992), 586-595.  doi: 10.1137/0405048.  Google Scholar  W. K. Hale, Frequency assignment: Theory and applications, Proc. IEEE, 68 (1980), 1497-1514.  doi: 10.1109/PROC.1980.11899. Google Scholar  R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, 2nd edition, CRC Press, Boca Raton, 2011. Google Scholar  P. K. Jha, S. Klavžar and A. Vesel, L(2, 1)-labeling of direct product of paths and cycles, Discrete. Appl. Math., 145 (2005), 317-325.  doi: 10.1016/j.dam.2004.01.019.  Google Scholar  S. Klavžar and A. Vesel, Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of (2, 1)-colorings and independence numbers, Discrete Appl. Math., 129 (2003), 449-460.  doi: 10.1016/S0166-218X(02)00597-8.  Google Scholar  D. Korže and A. Vesel, L(2, 1)-labeling of strong products of cycles, Inf. Process. Lett., 94 (2005), 183-190.  doi: 10.1016/j.ipl.2005.01.007.  Google Scholar  D. Král and R. Škrekovski, A theorem about channel assignment problem, SIAM J. Discrete Math., 16 (2003), 426-437.  doi: 10.1137/S0895480101399449.  Google Scholar  J. Kratochvíl, D. Kratsch and M. Liedloff, Exact algorithms for L(2, 1)-labeling of graphs, Proc. 32nd MFCS, 4708 (2007), 513-524. Google Scholar  M. Liang, X. Xu, J. Liang and Z. Shao, Upper bounds on the connection probability for 2-D meshes and tori, J. Parallel and Distrib. Comput., 72 (2012), 185-194.  doi: 10.1016/j.jpdc.2011.11.006. Google Scholar  S. Sen, 2-dipath and oriented L(2, 1)-labelings of some families of oriented planar graphs, Electronic Notes in Discrete Math., 38 (2011), 771-776.  doi: 10.1016/j.endm.2011.10.029. Google Scholar  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)-labelling for products of paths and cycles, IET Communications, 7 (2013), 715-720.   Google Scholar  J. J. Sylvester, Mathematical questions with their solutions, Educational Times, 41 (1884), 171-178.   Google Scholar  M. El-Zahar, S. Khamis and K. Nazzal, On the Domination number of the Cartesian product of the cycle of length n and any graph, Discrete Appl. Math., 155 (2007), 515-522.  doi: 10.1016/j.dam.2006.07.003.  Google Scholar  X. Zhang and J. Qian, L(p, q)-labeling and integer flow on planar graphs, The Computer Journal, 56 (2013), 785-792.   Google Scholar (a) Cartesian product of $\overrightarrow{P}_6$ and $\overrightarrow{P}_6$ (b) Cartesian product of $\overrightarrow{C}_6$ and $\overrightarrow{C}_6$ A 5- $L(2, 1)$-labeling of $\overrightarrow{C_{11}} \Box \overrightarrow{C_{11}}$ An 8- $L(2, 1)$-labeling of $\overrightarrow{C_{13}} \boxtimes \overrightarrow{C_{13}}$ An 8- $L(2, 1)$-labeling of $\overrightarrow{C_{5}} \boxtimes \overrightarrow{C_{13}}$ An 8- $L(2, 1)$-labeling of $\overrightarrow{C_{6}} \boxtimes \overrightarrow{C_{13}}$ An 8- $L(2, 1)$-labeling of $\overrightarrow{C_{7}} \boxtimes \overrightarrow{C_{23}}$ An 8- $L(2, 1)$-labeling of $\overrightarrow{C_{8}} \boxtimes \overrightarrow{C_{17}}$ An 8- $L(2, 1)$-labeling of $\overrightarrow{C_{9}} \boxtimes \overrightarrow{C_{17}}$ An 8- $L(2, 1)$-labeling of $\overrightarrow{C_{10}} \boxtimes \overrightarrow{C_{21}}$
Summary of results on $\lambda( \overrightarrow{C_m} \boxtimes \overrightarrow{C_n})$
 $m$ $k$ $|D_{m, k}|$ $\max\{d^+\}$ cycle lengths result 3 7 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 3 8 120 1 $\{9\}$ if $n \textrm{ mod }9 \equiv 0$, then $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 8$; otherwise $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 9$. 3 9 1800 4 ? $D_{3, 9}$ contains no closed walk of length from $\{3, 4, 5, 7, 8, 11, 14, 17\}$, thus $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 10$ for $n \in \{3, 4, 5, 7, 8, 11, 14, 17\}$. 4 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 4 7 72 1 $\{16\}$ if $n \equiv 0 \textrm{ mod } 16$, then $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 7$; otherwise $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 4 8 2664 5 ? $D_{4, 8}$ contains no closed walk of length from $S_4$, thus $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 9$ for $n \in S_4$. 5 7 40 1 $\emptyset$ $\lambda(\overrightarrow{{{C}_{5}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 5 8 10200 10 ? $D_{5, 8}$ contains no closed walk of length from $\{6, 7, 12\}$, thus $\lambda(\overrightarrow{{{C}_{5}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 9$ for $n \in \{6, 7, 12\}$. 6 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 6 7 540 4 $\{6\}$ if $n \equiv 0 \textrm{ mod } 6$, then $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 7$; otherwise $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 6 8 72534 27 ? $D_{6, 8}$ contains no closed walk of length 11, thus $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{11}}}) \geq 9$. 7 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{7}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 7 7 2296 8 ? $D_{7, 7}$ contains no closed walk of length from $S_7$, thus $\lambda(\overrightarrow{{{C}_{7}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$ for $n \in S_7$. 8 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{8}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. Continued on next page 8 7 720 1 $\{8, 16\}$ $n \equiv 0 \textrm{ mod } 8$, then $\lambda(\overrightarrow{{{C}_{8}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 7$; otherwise $\lambda(\overrightarrow{{{C}_{8}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 9 7 1530 2 $\emptyset$ $\lambda(\overrightarrow{{{C}_{9}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 10 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{10}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 10 7 16100 6 ? $D_{10, 7}$ contains no closed walk of length from $S_{10}$, thus $\lambda(\overrightarrow{{{C}_{10}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$ for $n \in S_{10}$.
 $m$ $k$ $|D_{m, k}|$ $\max\{d^+\}$ cycle lengths result 3 7 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 3 8 120 1 $\{9\}$ if $n \textrm{ mod }9 \equiv 0$, then $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 8$; otherwise $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 9$. 3 9 1800 4 ? $D_{3, 9}$ contains no closed walk of length from $\{3, 4, 5, 7, 8, 11, 14, 17\}$, thus $\lambda(\overrightarrow{{{C}_{3}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 10$ for $n \in \{3, 4, 5, 7, 8, 11, 14, 17\}$. 4 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 4 7 72 1 $\{16\}$ if $n \equiv 0 \textrm{ mod } 16$, then $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 7$; otherwise $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 4 8 2664 5 ? $D_{4, 8}$ contains no closed walk of length from $S_4$, thus $\lambda(\overrightarrow{{{C}_{4}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 9$ for $n \in S_4$. 5 7 40 1 $\emptyset$ $\lambda(\overrightarrow{{{C}_{5}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 5 8 10200 10 ? $D_{5, 8}$ contains no closed walk of length from $\{6, 7, 12\}$, thus $\lambda(\overrightarrow{{{C}_{5}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 9$ for $n \in \{6, 7, 12\}$. 6 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 6 7 540 4 $\{6\}$ if $n \equiv 0 \textrm{ mod } 6$, then $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 7$; otherwise $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 6 8 72534 27 ? $D_{6, 8}$ contains no closed walk of length 11, thus $\lambda(\overrightarrow{{{C}_{6}}}\boxtimes \overrightarrow{{{C}_{11}}}) \geq 9$. 7 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{7}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 7 7 2296 8 ? $D_{7, 7}$ contains no closed walk of length from $S_7$, thus $\lambda(\overrightarrow{{{C}_{7}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$ for $n \in S_7$. 8 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{8}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. Continued on next page 8 7 720 1 $\{8, 16\}$ $n \equiv 0 \textrm{ mod } 8$, then $\lambda(\overrightarrow{{{C}_{8}}}\boxtimes \overrightarrow{{{C}_{n}}}) \leq 7$; otherwise $\lambda(\overrightarrow{{{C}_{8}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 9 7 1530 2 $\emptyset$ $\lambda(\overrightarrow{{{C}_{9}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$. 10 6 0 0 $\emptyset$ $\lambda(\overrightarrow{{{C}_{10}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 7$. 10 7 16100 6 ? $D_{10, 7}$ contains no closed walk of length from $S_{10}$, thus $\lambda(\overrightarrow{{{C}_{10}}}\boxtimes \overrightarrow{{{C}_{n}}}) \geq 8$ for $n \in S_{10}$.
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