论文标题
最佳$ L（1,2）$ - 无限八角形网格的边缘标签
Optimal $L(1,2)$-edge Labeling of Infinite Octagonal Grid
论文作者
论文摘要
For two given non-negative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G=(V(G),E(G))$ is a function $f':E(G) \xrightarrow{}\{0,1,\cdots, n\}$ such that $\forall e_1,e_2 \in E(G)$, $ \ vert f'（e_1）-f'（e_2）\ vert \ geq h $时$ d'（e_1，e_2）= 1 $ and $ \ vert f'（e_1）-f'（e_2）\ vert \ geq k $时和$ e_2 $ in $ g $。这里$ d'（e_1，e_2）= k'$，如果至少有$（k'-1）$ e（g）$中的边缘数（g）$连接$ e_1 $和$ g $中的$ e_2 $。目的是找到\ textit {span}，这是所有此类$ l（h，k）$ - 边缘标签的最小$ n $，并表示为$λ'__ {h，k}（g）$。由无线蜂窝网络中的频道分配问题激励，$ L（H，K）$ - 边缘标签问题已在各种无限的常规网格中进行了研究。对于无限的常规八角形电网$ T_8 $，证明$ 25 \leqλ'_{1,2}（T_8）\ leq 28 $ [Tiziana Calamoneri，国际计算机科学基础杂志，第1卷。 26，第04号，2015年]，下限和上限之间有差距。在本文中，我们填补了空白，并证明$λ'_{1,2}（T_8）= 28 $。
For two given non-negative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G=(V(G),E(G))$ is a function $f':E(G) \xrightarrow{}\{0,1,\cdots, n\}$ such that $\forall e_1,e_2 \in E(G)$, $\vert f'(e_1)-f'(e_2) \vert \geq h$ when $d'(e_1,e_2)=1$ and $\vert f'(e_1)-f'(e_2) \vert \geq k$ when $d'(e_1,e_2)=2$ where $d'(e_1,e_2)$ denotes the distance between $e_1$ and $e_2$ in $G$. Here $d'(e_1,e_2)=k'$ if there are at least $(k'-1)$ number of edges in $E(G)$ to connect $e_1$ and $e_2$ in $G$. The objective is to find \textit{span} which is the minimum $n$ over all such $L(h,k)$-edge labeling and is denoted as $λ'_{h,k}(G)$. Motivated by the channel assignment problem in wireless cellular network, $L(h,k)$-edge labeling problem has been studied in various infinite regular grids. For infinite regular octagonal grid $T_8$, it was proved that $25 \leq λ'_{1,2}(T_8) \leq 28$ [Tiziana Calamoneri, International Journal of Foundations of Computer Science, Vol. 26, No. 04, 2015] with a gap between lower and upper bounds. In this paper we fill the gap and prove that $λ'_{1,2}(T_8)= 28$.
