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

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

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  • 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 $m, n ≥ 48$.

    Mathematics Subject Classification: Primary: 05C90, 05C85.

    Citation:

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  • Figure 1.  (a) Cartesian product of $ \overrightarrow{P}_6$ and $ \overrightarrow{P}_6$ (b) Cartesian product of $ \overrightarrow{C}_6$ and $ \overrightarrow{C}_6$

    Figure 2.  A 5- $L(2, 1)$-labeling of $ \overrightarrow{C_{11}} \Box \overrightarrow{C_{11}}$

    Figure 3.  An 8- $L(2, 1)$-labeling of $ \overrightarrow{C_{13}} \boxtimes \overrightarrow{C_{13}}$

    Figure 4.  An 8- $L(2, 1)$-labeling of $ \overrightarrow{C_{5}} \boxtimes \overrightarrow{C_{13}}$

    Figure 5.  An 8- $L(2, 1)$-labeling of $ \overrightarrow{C_{6}} \boxtimes \overrightarrow{C_{13}}$

    Figure 6.  An 8- $L(2, 1)$-labeling of $ \overrightarrow{C_{7}} \boxtimes \overrightarrow{C_{23}}$

    Figure 7.  An 8- $L(2, 1)$-labeling of $ \overrightarrow{C_{8}} \boxtimes \overrightarrow{C_{17}}$

    Figure 8.  An 8- $L(2, 1)$-labeling of $ \overrightarrow{C_{9}} \boxtimes \overrightarrow{C_{17}}$

    Figure 9.  An 8- $L(2, 1)$-labeling of $ \overrightarrow{C_{10}} \boxtimes \overrightarrow{C_{21}}$

    Table 1.  Summary of results on $\lambda( \overrightarrow{C_m} \boxtimes \overrightarrow{C_n})$

    $m$ $k$ $|D_{m, k}|$ $\max\{d^+\}$cycle lengthsresult
    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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