论文标题
在$ L（1,2）$ - 边缘标签的某些无限常规网格的跨度上有改进的界限
Improved Bounds on the Span of $L(1,2)$-edge Labeling of Some Infinite Regular Grids
论文作者
论文摘要
For two given nonnegative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G$ is the assignment of labels $\{0,1, \cdots, n\}$ to the edges so that two edges having a common vertex are labeled with difference at least $h$ and two edges not having any common vertex but having a common edge connecting them are labeled with difference at least $ k $。跨度$λ'_ {h，k} {（g）} $是最低$ n $，因此$ g $允许$ l（h，k）$ - 边缘标签。在这里，我们的主要重点是寻找$λ'_ {1,2} {（g）} $，用于$ l（1,2）$ - 无限常规六角形（$ t_3 $），square（$ t_4 $），三角形（$ t_6 $）和八尖塔（$ t_8 $）的网格。众所周知，$ 7 \leqλ'_{1,2} {（t_3）} \ leq 8 $，$ 10 \leqλ'__ {1,2} {（t_4）} \ leq 11 $，$ 16 \ leqleqλ'__{1,2} \ 25 λ'_{1,2} {（T_8）} \ leq 28 $。在这里，我们解决两个长期的开放问题，即$λ'_{1,2} {（T_3）} $和$λ'__ {1,2} {（T_4）} $。我们显示$λ'_ {1,2} {（t_3）} = 7 $，$λ'_{1,2} {（t_4）} = 11 $。我们还改善了$ T_6 $和$ T_8 $的界限，并证明$λ'_ {1,2} {（T_6）} \ geq 18 $，$λ'_ {1,2} {（T_8）} {（t_8）} \ geq 26 $。
For two given nonnegative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G$ is the assignment of labels $\{0,1, \cdots, n\}$ to the edges so that two edges having a common vertex are labeled with difference at least $h$ and two edges not having any common vertex but having a common edge connecting them are labeled with difference at least $k$. The span $λ'_{h,k}{(G)}$ is the minimum $n$ such that $G$ admits an $L(h,k)$-edge labeling. Here our main focus is on finding $λ'_{1,2}{(G)}$ for $L(1,2)$-edge labeling of infinite regular hexagonal ($T_3$), square ($T_4$), triangular ($T_6$) and octagonal ($T_8$) grids. It was known that $7 \leq λ'_{1,2}{(T_3)} \leq 8$, $10 \leq λ'_{1,2}{(T_4)} \leq 11$, $16 \leq λ'_{1,2}{(T_6)} \leq 20$ and $25 \leq λ'_{1,2}{(T_8)} \leq 28$. Here we settle two long standing open questions i.e. $λ'_{1,2}{(T_3)}$ and $λ'_{1,2}{(T_4)}$. We show $λ'_{1,2}{(T_3)} =7$, $λ'_{1,2}{(T_4)}= 11$. We also improve the bound for $T_6$ and $T_8$ and prove $λ'_{1,2}{(T_6)} \geq 18$, $ λ'_{1,2}{(T_8)} \geq 26$.
